text: 4.5 Bio-Inspired Neural Networks 4 NUMERICAL EXPERIMENTS ± Diehl and Cook [2015]. Using self-labeling instead, ff+-NeSi achieved a test error of (5.10 0.18)%. The semi-supervised settings have not been investigated by Diehl and Cook [2015] but it would represent interesting data for comparison, and it should be straightforward to operate the spiking network model also in this regime. The second system we compare to is the Synaptic Sampling Machine (SSM), recently suggested by Neftci × et al. [2015]. The network consist of 28 28 = 784 neurons in the input layer, 500 neurons in the ﬁrst hidden layer and 10 neurons in the top layer. Inference and learning is implemented based on spiking neurons with MNIST data represented by Poisson spike trains. The SSM is closely related to Restricted Boltzmann Machines [RBMs; see, for example, Dayan and Abbott, 2001, Salakhutdinov and Hinton, 2009] with weight changes following a continuous time variant of contrastive divergence [Hinton, 2002]. If trained on the MNIST training set using all labels, the SSM achieves an error rate of 4.4% on the test set. In addition to the SSM, Neftci et al. [2015] also considered standard discrete time RBMs with the same architecture (784–500–10). Using the most conventional setting with Gibbs sampling and standard contrastive divergence, the RBM obtained an optimal test error of 5.0% (again using all labels). Learning for the RBM was here assumed to stop at the point of optimal performance while longer learning resulted in larger test errors which were attributed to decreased MCMC ergodicity and overﬁtting [Neftci et al., 2015]. An improved RBM variant, the dSSM network, did not suffer from such overtraining effects. The test error of the dSSM (architecture 784–500–10) on fully labeled MNIST was 4.5%. The SSM, RBM and dSSM systems have essentially the same network architecture as our NeSi systems if we use 500 middle layer neurons. For this setting (without further optimization of the remaining free parameters), ± ± the ff-NeSi network achieved a test error of (4.95 0.03)% and r-NeSi of (3.97 0.03)% (both fully ± labeled). When using only 100 labels for training, r+-NeSi achieved a test error of (11.50 1.37)% and ± ff+-NeSi of (8.36 0.64)%. So far, we used the usual training setup, where classes are learned from labeled training data. An alternative evaluation procedure is suggested by Nessler et al. [2013] for the ﬁnal system we compare to. The spike-based Expectation Maximization approach (SEM) implements a generative Poisson model as spiking neural network with STDP rules. The network is trained unsupervised with 100 neurons on MNIST. The class assignment of the learned representations is then done directly by the user who inspects the ﬁelds and assigns to each ﬁeld what he considers the most likely label. With this procedure, the network of Nessler et al. [2013] achieves a test error of 19.86%. We can adopt the same procedure by only training the ﬁrst layer of an ff-NeSi network and then assigning the ﬁelds manually with labels l(c) by setting the weights R to kc (cid:80) δ R = l(c)k . (24) kc (cid:48) δ (cid:48) c l(c )k ± Using this procedure, the NeSi network achieved a test error of (10.53 0.11)%. For functional goals, we could further improve on these results. We could, for example, ask the user to assign a certainty weight to the chosen labels or even ask to assign a probability distribution over all possible labels. Improvements are also possible without requesting further information from the supervisor. By using recurrence and self-labeling of the r+-NeSi network, we were able to improve classiﬁcation down to an error rate of ± (5.15 0.26)% based on 100 ﬁelds labeled by the user (see ?? for details). Notably, other lines of research [for example, Esser et al., 2015, Diehl et al., 2015] do not consider networks for spike-based learning and inference but focus on spike-based inference alone. Typically, in a ﬁrst stage, standard (non-spiking) discriminative networks are trained using conventional back- 24 ##LTLine## ##LTLine##

text: Bio-Inspired

text: Neural

text: Networks

text: NUMERICAL

text: EXPERIMENTS

text: Diehl

text: and

text: Cook

text: Using

text: self-labeling

text: instead,

text: ff+-NeSi

text: achieved

text: a

text: test

text: error

text: of

text: The

text: semi-supervised

text: settings

text: have

text: not

text: been

text: investigated

text: by

text: Diehl

text: and

text: Cook

text: but

text: it

text: would

text: represent

text: interesting

text: data

text: for

text: comparison,

text: and

text: it

text: should

text: be

text: straightforward

text: to

text: operate

text: the

text: spiking

text: network

text: model

text: also

text: in

text: this

text: regime.

text: The

text: second

text: system

text: we

text: compare

text: to

text: is

text: the

text: Synaptic

text: Sampling

text: Machine

text: (SSM),

text: recently

text: suggested

text: by

text: Neftci

text: et

text: al.

text: The

text: network

text: consist

text: of

text: neurons

text: in

text: the

text: input

text: layer,

text: neurons

text: in

text: the

text: ﬁrst

text: hidden

text: layer

text: and

text: neurons

text: in

text: the

text: top

text: layer.

text: Inference

text: and

text: learning

text: is

text: implemented

text: based

text: on

text: spiking

text: neurons

text: with

text: MNIST

text: data

text: represented

text: by

text: Poisson

text: spike

text: trains.

text: The

text: SSM

text: is

text: closely

text: related

text: to

text: Restricted

text: Boltzmann

text: Machines

text: [RBMs;

text: see,

text: for

text: example,

text: Dayan

text: and

text: Abbott,

text: Salakhutdinov

text: and

text: Hinton,

text: with

text: weight

text: changes

text: following

text: a

text: continuous

text: time

text: variant

text: of

text: contrastive

text: divergence

text: [Hinton,

text: If

text: trained

text: on

text: the

text: MNIST

text: training

text: set

text: using

text: all

text: labels,

text: the

text: SSM

text: achieves

text: an

text: error

text: rate

text: of

text: on

text: the

text: test

text: set.

text: In

text: addition

text: to

text: the

text: SSM,

text: Neftci

text: et

text: al.

text: also

text: considered

text: standard

text: discrete

text: time

text: RBMs

text: with

text: the

text: same

text: architecture

text: Using

text: the

text: most

text: conventional

text: setting

text: with

text: Gibbs

text: sampling

text: and

text: standard

text: contrastive

text: divergence,

text: the

text: RBM

text: obtained

text: an

text: optimal

text: test

text: error

text: of

text: (again

text: using

text: all

text: labels).

text: Learning

text: for

text: the

text: RBM

text: was

text: here

text: assumed

text: to

text: stop

text: at

text: the

text: point

text: of

text: optimal

text: performance

text: while

text: longer

text: learning

text: resulted

text: in

text: larger

text: test

text: errors

text: which

text: were

text: attributed

text: to

text: decreased

text: MCMC

text: ergodicity

text: and

text: overﬁtting

text: [Neftci

text: et

text: al.,

text: An

text: improved

text: RBM

text: variant,

text: the

text: dSSM

text: network,

text: did

text: not

text: suffer

text: from

text: such

text: overtraining

text: effects.

text: The

text: test

text: error

text: of

text: the

text: dSSM

text: (architecture

text: on

text: fully

text: labeled

text: MNIST

text: was

text: The

text: SSM,

text: RBM

text: and

text: dSSM

text: systems

text: have

text: essentially

text: the

text: same

text: network

text: architecture

text: as

text: our

text: NeSi

text: systems

text: if

text: we

text: use

text: middle

text: layer

text: neurons.

text: For

text: this

text: setting

text: (without

text: further

text: optimization

text: of

text: the

text: remaining

text: free

text: parameters),

text: the

text: ff-NeSi

text: network

text: achieved

text: a

text: test

text: error

text: of

text: and

text: r-NeSi

text: of

text: (both

text: fully

text: labeled).

text: When

text: using

text: only

text: labels

text: for

text: training,

text: r+-NeSi

text: achieved

text: a

text: test

text: error

text: of

text: and

text: ff+-NeSi

text: of

text: So

text: far,

text: we

text: used

text: the

text: usual

text: training

text: setup,

text: where

text: classes

text: are

text: learned

text: from

text: labeled

text: training

text: data.

text: An

text: alternative

text: evaluation

text: procedure

text: is

text: suggested

text: by

text: Nessler

text: et

text: al.

text: for

text: the

text: ﬁnal

text: system

text: we

text: compare

text: to.

text: The

text: spike-based

text: Expectation

text: Maximization

text: approach

text: (SEM)

text: implements

text: a

text: generative

text: Poisson

text: model

text: as

text: spiking

text: neural

text: network

text: with

text: STDP

text: rules.

text: The

text: network

text: is

text: trained

text: unsupervised

text: with

text: neurons

text: on

text: MNIST.

text: The

text: class

text: assignment

text: of

text: the

text: learned

text: representations

text: is

text: then

text: done

text: directly

text: by

text: the

text: user

text: who

text: inspects

text: the

text: ﬁelds

text: and

text: assigns

text: to

text: each

text: ﬁeld

text: what

text: he

text: considers

text: the

text: most

text: likely

text: label.

text: With

text: this

text: procedure,

text: the

text: network

text: of

text: Nessler

text: et

text: al.

text: achieves

text: a

text: test

text: error

text: of

text: We

text: can

text: adopt

text: the

text: same

text: procedure

text: by

text: only

text: training

text: the

text: ﬁrst

text: layer

text: of

text: an

text: ff-NeSi

text: network

text: and

text: then

text: assigning

text: the

text: ﬁelds

text: manually

text: with

text: labels

text: l(c)

text: by

text: setting

text: the

text: weights

text: R

text: to

text: kc

text: δ

text: R

text: l(c)k

text: kc

text: δ

text: c

text: l(c

text: )k

text: Using

text: this

text: procedure,

text: the

text: NeSi

text: network

text: achieved

text: a

text: test

text: error

text: of

text: For

text: functional

text: goals,

text: we

text: could

text: further

text: improve

text: on

text: these

text: results.

text: We

text: could,

text: for

text: example,

text: ask

text: the

text: user

text: to

text: assign

text: a

text: certainty

text: weight

text: to

text: the

text: chosen

text: labels

text: or

text: even

text: ask

text: to

text: assign

text: a

text: probability

text: distribution

text: over

text: all

text: possible

text: labels.

text: Improvements

text: are

text: also

text: possible

text: without

text: requesting

text: further

text: information

text: from

text: the

text: supervisor.

text: By

text: using

text: recurrence

text: and

text: self-labeling

text: of

text: the

text: r+-NeSi

text: network,

text: we

text: were

text: able

text: to

text: improve

text: classiﬁcation

text: down

text: to

text: an

text: error

text: rate

text: of

text: based

text: on

text: ﬁelds

text: labeled

text: by

text: the

text: user

text: (see

text: for

text: details).

text: Notably,

text: other

text: lines

text: of

text: research

text: [for

text: example,

text: Esser

text: et

text: al.,

text: Diehl

text: et

text: al.,

text: do

text: not

text: consider

text: networks

text: for

text: spike-based

text: learning

text: and

text: inference

text: but

text: focus

text: on

text: spike-based

text: inference

text: alone.

text: Typically,

text: in

text: a

text: ﬁrst

text: stage,

text: standard

text: (non-spiking)

text: discriminative

text: networks

text: are

text: trained

text: using

text: conventional

text: back-

text: ##LTLine##

text: ##LTLine##


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text: Joshi,

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text: Multi-class

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text: Computer

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text: Recognition

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text: J.

text: Lücke.

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text: two

text: sides

text: of

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text: same

text: coin?

text: PLoS

text: Computational

text: Biology,

text: D.-k.

text: Kim,

text: M.

text: Der,

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text: K.

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text: A

text: Gaussian

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text: Conference

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text: Learning

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text: pages

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text: Larochelle

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text: Y.

text: Bengio.

text: Classiﬁcation

text: using

text: discriminative

text: restricted

text: Boltzmann

text: machines.

text: In

text: International

text: Conference

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text: Learning

text: (ICML),

text: pages

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text: and

text: I.

text: Murray.

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text: neural

text: autoregressive

text: distribution

text: estimator.

text: In

text: International

text: Conference

text: on

text: Artiﬁcial

text: Intelligence

text: and

text: Statistics

text: (AISTATS),

text: pages

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text: L.

text: Bottou,

text: Y.

text: Bengio,

text: and

text: P.

text: Haffner.

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text: learning

text: applied

text: to

text: document

text: recognition.

text: IEEE,

text: D.-H.

text: Lee.

text: Pseudo-label:

text: The

text: simple

text: and

text: efﬁcient

text: semi-supervised

text: learning

text: method

text: for

text: deep

text: neural

text: networks.

text: In

text: Workshop

text: on

text: Challenges

text: in

text: Representation

text: Learning,

text: ICML,

text: volume

text: page

text: T.

text: S.

text: Lee

text: and

text: D.

text: Mumford.

text: Hierarchical

text: Bayesian

text: inference

text: in

text: the

text: visual

text: cortex.

text: JOSA

text: A,

text: W.

text: Liu,

text: J.

text: He,

text: and

text: S.-F.

text: Chang.

text: Large

text: graph

text: construction

text: for

text: scalable

text: semi-supervised

text: learning.

text: In

text: International

text: Conference

text: on

text: Machine

text: Learning

text: (ICML),

text: pages

text: J.

text: Lücke

text: and

text: M.

text: Sahani.

text: Generalized

text: softmax

text: networks

text: for

text: non-linear

text: component

text: extraction.

text: In

text: International

text: Conference

text: on

text: Artiﬁcial

text: Neural

text: Networks

text: (ICANN),

text: LNCS

text: pages

text: Springer,

text: J.

text: Lücke

text: and

text: M.

text: Sahani.

text: Maximal

text: causes

text: for

text: non-linear

text: component

text: extraction.

text: Journal

text: of

text: Machine

text: Learning

text: Research

text: (JMLR),

text: T.

text: Miyato,

text: S.-i.

text: Maeda,

text: M.

text: Koyama,

text: K.

text: Nakae,

text: and

text: S.

text: Ishii.

text: Distributional

text: smoothing

text: with

text: virtual

text: adversarial

text: training.

text: arXiv

text: preprint

text: S.

text: Mohamed,

text: K.

text: A.

text: Heller,

text: and

text: Z.

text: Ghahramani.

text: Evaluating

text: Bayesian

text: and

text: approaches

text: for

text: sparse

text: unsupervised

text: learning.

text: In

text: International

text: Conference

text: on

text: Machine

text: Learning

text: (ICML).

text: icml.

text: cc/Omnipress,

text: R.

text: M.

text: Neal

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text: G.

text: E.

text: Hinton.

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text: of

text: the

text: EM

text: algorithm

text: that

text: justiﬁes

text: incremental,

text: sparse,

text: and

text: other

text: variants.

text: In

text: Learning

text: in

text: Graphical

text: Models,

text: pages

text: Springer,

text: E.

text: O.

text: Neftci,

text: B.

text: U.

text: Pedroni,

text: S.

text: Joshi,

text: M.

text: Al-Shedivat,

text: and

text: G.

text: Cauwenberghs.

text: Unsupervised

text: learning

text: in

text: Synaptic

text: Sampling

text: Machines.

text: arXiv

text: preprint

text: ##LTLine##


text: and each term is either a continuous and bounded functional, or its convergence is addressed directly by (cid:48) → ∈ (cid:48) the construction of Ω , we have P (ω)g P g as required. As ω Ω was arbitrary, the result holds n ν ν almost surely. Remark 3.5. Note that in the above theorem we did not need a lower bound on the decay of λ (only ≥ n that λ 0). We only used that λ = o(1). n n 3.2 Uniqueness of the -limit Γ To show the Γ-limit has a unique minimizer we show it is strictly convex. The following lemma gives the second Gaˆteaux derivative of f∞. After which we conclude in Corollary 3.7 that the Γ-limit is unique. Lemma 3.6. Under Assumptions 1-2 de(cid:90)ﬁn(cid:90)e f∞ : H → R by (6). Then the ﬁrst and second Gaˆteaux derivatives of f∞ are given by ∞ (cid:90) − ∂f∞(µ; ν) = 2 (L µ y)L (ν)φ †(dy)φ (dt) I −∞ t t Ltµ T ∂2f∞(µ; ν, ζ) = 2 (L ν)(L ζ)φ (dt). I t t T (cid:90) (cid:90) Proof. We ﬁrst compute the ﬁrst Gaˆteaux derivative. We have ∞ − − − (cid:90) (cid:90) (y L (µ + rν))2 (y L µ)2 (cid:90) (cid:90) ∂f∞(µ; ν) = lim t t φ †(dy)φ (dt) r→0 I −∞ r Ltµ T ∞ ∞ (cid:90) (cid:90) − = 2 (L µ y)L (ν)φ †(dy)φ (dt) + lim r (L ν)2φ †(dy)φ (dt) I −∞ t t Ltµ T r→0 I −∞ t Ltµ T ∞ − = 2 (L µ y)L (ν)φ †(dy)φ (dt) recalling that L is linear. I −∞ t t Ltµ T t (cid:90) (cid:90) The second Gaˆteaux derivative follows similarly. ∞ − − − (cid:90) (cid:90) (L (µ + rζ) y)L ν (L µ y)L ν ∂2f∞(µ; ν, ζ) = lim 2 t t t t φ †(dy)φ (dt) r→0 I −∞ r Ltµ T ∞ (cid:90) = 2 (L ν)(L ζ)φ †(dy)φ (dt) I −∞ t t Ltµ T = 2 (L ν)(L ζ)φ (dt). I t t T Corollary 3.7. Under Assumptions 1-2 and 6, deﬁne f∞ : H → R by (6). Then f∞ has a unique † minimizer which is achieved for µ = µ . † ∈ H Proof. It is easy to check that ∂f∞(µ ; ν) = 0 for all ν . By Lemma 3.6 and Assumption 6 (cid:54) the second Gaˆteaux derivative satisﬁes ∂2f∞(µ; ν) > 0 for all ν = 0. Then by Taylor’s Theorem (and (cid:54) † noting that f∞ is quadratic), for µ = µ , † 1 † − † † f∞(µ) = f∞(µ ) + ∂2f∞(µ ; µ µ ) > f∞(µ ) 2 as required. 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: each

text: term

text: is

text: either

text: a

text: continuous

text: and

text: bounded

text: functional,

text: or

text: its

text: convergence

text: is

text: addressed

text: directly

text: by

text: the

text: construction

text: of

text: Ω

text: we

text: have

text: P

text: (ω)g

text: P

text: g

text: as

text: required.

text: As

text: ω

text: Ω

text: was

text: arbitrary,

text: the

text: result

text: holds

text: n

text: ν

text: ν

text: almost

text: surely.

text: Remark

text: Note

text: that

text: in

text: the

text: above

text: theorem

text: we

text: did

text: not

text: need

text: a

text: lower

text: bound

text: on

text: the

text: decay

text: of

text: λ

text: (only

text: n

text: that

text: λ

text: We

text: only

text: used

text: that

text: λ

text: n

text: n

label: section

text: Uniqueness

text: of

text: the

text: -limit

text: Γ

text: To

text: show

text: the

text: Γ-limit

text: has

text: a

text: unique

text: minimizer

text: we

text: show

text: it

text: is

text: strictly

text: convex.

text: The

text: following

text: lemma

text: gives

text: the

text: second

text: Gaˆteaux

text: derivative

text: of

text: f∞.

text: After

text: which

text: we

text: conclude

text: in

text: Corollary

text: that

text: the

text: Γ-limit

text: is

text: unique.

text: Lemma

text: Under

text: Assumptions

text: f∞

text: H

text: R

text: by

text: Then

text: the

text: ﬁrst

text: and

text: second

text: Gaˆteaux

text: derivatives

text: of

text: f∞

text: are

text: given

text: by

text: ∂f∞(µ;

text: ν)

text: (L

text: µ

text: y)L

text: (ν)φ

text: †(dy)φ

text: (dt)

text: I

text: t

text: t

text: Ltµ

text: T

text: ν,

text: ζ)

text: (L

text: ν)(L

text: ζ)φ

text: (dt).

text: I

text: t

text: t

text: T

text: Proof.

text: We

text: ﬁrst

text: compute

text: the

text: ﬁrst

text: Gaˆteaux

text: derivative.

text: We

text: have

text: (y

text: L

text: (µ

text: (y

text: L

text: ∂f∞(µ;

text: ν)

text: lim

text: t

text: t

text: φ

text: †(dy)φ

text: (dt)

text: I

text: r

text: Ltµ

text: T

text: (L

text: µ

text: y)L

text: (ν)φ

text: †(dy)φ

text: (dt)

text: lim

text: r

text: (L

text: ν)2φ

text: †(dy)φ

text: (dt)

text: I

text: t

text: t

text: Ltµ

text: T

text: I

text: t

text: Ltµ

text: T

text: (L

text: µ

text: y)L

text: (ν)φ

text: †(dy)φ

text: (dt)

text: recalling

text: that

text: L

text: is

text: linear.

text: I

text: t

text: t

text: Ltµ

text: T

text: t

text: The

text: second

text: Gaˆteaux

text: derivative

text: follows

text: similarly.

text: (L

text: (µ

text: rζ)

text: y)L

text: ν

text: (L

text: µ

text: y)L

text: ν

text: ν,

text: ζ)

text: lim

text: t

text: t

text: t

text: t

text: φ

text: †(dy)φ

text: (dt)

text: I

text: r

text: Ltµ

text: T

text: (L

text: ν)(L

text: ζ)φ

text: †(dy)φ

text: (dt)

text: I

text: t

text: t

text: Ltµ

text: T

text: (L

text: ν)(L

text: ζ)φ

text: (dt).

text: I

text: t

text: t

text: T

text: Corollary

text: Under

text: Assumptions

text: and

text: deﬁne

text: f∞

text: H

text: R

text: by

text: Then

text: f∞

text: has

text: a

text: unique

text: minimizer

text: which

text: is

text: achieved

text: for

text: µ

text: µ

text: H

text: Proof.

text: It

text: is

text: easy

text: to

text: check

text: that

text: ∂f∞(µ

text: ν)

text: for

text: all

text: ν

text: By

text: Lemma

text: and

text: Assumption

text: the

text: second

text: Gaˆteaux

text: derivative

text: satisﬁes

text: ν)

text: for

text: all

text: ν

text: Then

text: by

text: Taylor’s

text: Theorem

text: (and

text: noting

text: that

text: f∞

text: is

text: quadratic),

text: for

text: µ

text: µ

text: f∞(µ)

text: f∞(µ

text: ∂2f∞(µ

text: µ

text: µ

text: f∞(µ

text: as

text: required.

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text: (cid:80) almost surely. Now, deﬁne νn =(cid:20)(cid:13)(cid:13)(cid:13)n1 ni=1 (cid:15)i(cid:13)(cid:13)(cid:13)ηi(cid:12)(cid:12)(cid:12)(cid:12)and(cid:21) (cid:88) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) E − 2 G a.s. σ2 n − 2 G 1 ν = G 1 η n,λn n 1 n n2 n,λn i 1 i=1 ≤ α2σ2 . nλ2 n H Combined with Lemma 3.8 (the bound) this proves the lemma. 0 Recalling (8) and via Lemmas 3.11 and 3.12 we obtain the following asymptotic bound on minimiz- H ers in . (cid:2) (cid:3) Theorem 3.13. Under Assumptions 1-5 we have E (cid:107) (cid:107) |G µn 2 = O(1) almost surely. (10) n (cid:107) (cid:107) This is a stronger result than we needed; we were only required to show that µn is bounded in probability. Taking expectation of (10) one has E(cid:107) (cid:107) µn 2 = O(1). (cid:107) (cid:107) Hence applying Chebyshev’s inequality we may conclude that µn = O (1). p (cid:107) (cid:107) Corollary 3.14. Under Assumptions 1-5 we have µn = O (1). p ∈ H∗ We conclude this section with a brief analysis of the rate of convergence. For any F , by the ∈ H ∈ H Riesz Representation Theorem, there exists ξ such that F (µ) = (µ, ξ) for all µ . Hence (cid:80) − † − − † − F (µn) F (µ ) = ((G 1 U Id)µ + G 1 νn, ξ) n,λn n n,λn H H ⊕ ⊥ where νn = 1 n (cid:15)(cid:16)η(cid:16). Decomposing int(cid:17)o =(cid:17)Ran(cid:16)(U ) Ran(U (cid:17)) o(cid:16)ne can write(cid:17) n i=1 i i n n − † (cid:88) − − (cid:16) (cid:17) (cid:16)† − (cid:17) (cid:16) † (cid:17) −(cid:16) (cid:17) F (µn) F (µ ) = G 1 U χ µ , ξ χ ⊥µ , ξ + G 1 νn, ξ n,λn n Ran(Un) Ran(Un) n,λn − n λ † − † − = n µ , ψ(n) ψ(n), ξ χ ⊥µ , ξ + G 1 νn, ξ (11) β(n) + λ j j Ran(Un) n,λn j=1 j n where χ is the projection onto Ran(U(cid:88)). If we assume Ran(Un) n n 1 ∞ lim < . (12) →∞ n β(n) (cid:88) (cid:16) (cid:17)j(cid:16)=1 j (cid:17) (cid:88) Then − n n λ † ≤ (cid:107) †(cid:107)(cid:107) (cid:107) 1 n (n) (n) µ , ψ ψ , ξ µ ξ λ . (n) j j n (n) β + λ β j=1 j n j=1 j − And therefore the ﬁrst term in (11) is of the order n p. By the proof of Lemma 3.12 the third term in (11) is of order √ 1 . The second term is independent of λ . The optimal rate of convergence is nλn n therefore found by balancing the ﬁrst and third terms. This will imply an optimal choice of p = 1. We 4 summarise in the following proposition. 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: almost

text: surely.

text: Now,

text: deﬁne

text: νn

text: E

text: G

text: a.s.

text: σ2

text: n

text: G

text: ν

text: G

text: η

text: n,λn

text: n

text: n

text: n,λn

text: i

text: α2σ2

text: n

text: H

text: Combined

text: with

text: Lemma

text: (the

text: bound)

text: this

text: proves

text: the

text: lemma.

text: Recalling

text: and

text: via

text: Lemmas

text: and

text: we

text: obtain

text: the

text: following

text: asymptotic

text: bound

text: on

text: minimiz-

text: H

text: ers

text: in

label: equation

text: Theorem

text: Under

text: Assumptions

text: we

text: have

text: E

text: |G

text: µn

text: almost

text: surely.

text: n

text: This

text: is

text: a

text: stronger

text: result

text: than

text: we

text: needed;

text: we

text: were

text: only

text: required

text: to

text: show

text: that

text: µn

text: is

text: bounded

text: in

text: probability.

text: Taking

text: expectation

text: of

text: one

text: has

text: µn

text: Hence

text: applying

text: Chebyshev’s

text: inequality

text: we

text: may

text: conclude

text: that

text: µn

text: O

text: p

text: Corollary

text: Under

text: Assumptions

text: we

text: have

text: µn

text: O

text: p

text: H∗

text: We

text: conclude

text: this

text: section

text: with

text: a

text: brief

text: analysis

text: of

text: the

text: rate

text: of

text: convergence.

text: For

text: any

text: F

text: by

text: the

text: H

text: H

text: Riesz

text: Representation

text: Theorem,

text: there

text: exists

text: ξ

text: such

text: that

text: F

text: (µ)

text: (µ,

text: ξ)

text: for

text: all

text: µ

text: Hence

text: F

text: (µn)

text: F

text: (µ

text: ((G

text: U

text: Id)µ

text: G

text: νn,

text: ξ)

text: n,λn

text: n

text: n,λn

text: H

text: H

text: where

text: νn

text: n

text: Decomposing

text: Ran(U

text: can

text: n

text: i

text: i

text: n

text: n

text: F

text: (µn)

text: F

text: (µ

text: G

text: U

text: χ

text: µ

text: ξ

text: χ

text: ⊥µ

text: ξ

text: G

text: νn,

text: ξ

text: n,λn

text: n

text: Ran(Un)

text: Ran(Un)

text: n,λn

text: n

text: λ

text: n

text: µ

text: ψ(n)

text: ψ(n),

text: ξ

text: χ

text: ⊥µ

text: ξ

text: G

text: νn,

text: ξ

text: β(n)

text: λ

text: j

text: j

text: Ran(Un)

text: n,λn

text: j

text: n

text: where

text: χ

text: is

text: the

text: projection

text: onto

text: If

text: we

text: assume

text: Ran(Un)

text: n

text: n

text: lim

text: n

text: β(n)

text: j

text: Then

text: n

text: n

text: λ

text: n

text: (n)

text: (n)

text: µ

text: ψ

text: ψ

text: ξ

text: µ

text: ξ

text: λ

text: (n)

text: j

text: j

text: n

text: (n)

text: β

text: λ

text: β

text: j

text: n

text: j

text: And

text: therefore

text: the

text: ﬁrst

text: term

text: in

text: is

text: of

text: the

text: order

text: n

text: p.

text: By

text: the

text: proof

text: of

text: Lemma

text: the

text: third

text: term

text: in

text: is

text: of

text: order

text: The

text: second

text: term

text: is

text: independent

text: of

text: λ

text: The

text: optimal

text: rate

text: of

text: convergence

text: is

text: nλn

text: n

text: therefore

text: found

text: by

text: balancing

text: the

text: ﬁrst

text: and

text: third

text: terms.

text: This

text: will

text: imply

text: an

text: optimal

text: choice

text: of

text: p

text: We

text: summarise

text: in

text: the

text: following

text: proposition.

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text: ##LTLine##


text: (cid:2) (cid:3) particular for scaling outside this regime we construct (uniformly bounded) observation operators L E (cid:107) (cid:107) → ∞ i such that µn 2 . Hence without further assumptions our results are sharp. † We note that these results have practical implications. If we are interested in estimating µ at a point ∈ H∗ t then we let F (µ) = µ(t) where F . In this setting weak convergence, or the pointwise form considered in this paper, are the natural modes of convergence to consider. Whereas, if one is interested † − † in a global approximation of µ , then convergence of µn µ in an appropriate norm is the more relevant. The two formulations imply different scaling results for λ . n There are many results in the ill-posed inverse problems literature that may be applied to the strong convergence of the general spline problem, for brevity we only mention those most relevant to this work. → ∞ In [43] two different methods of estimating λ were compared as n using the general spline n (cid:80) formulation. The reproducing kernel Hilbert space setting was used in [21] which also discussed the probabilistic interpretation behind the estimator µn. In [11,30] the authors prove the strong convergence ∗ ≈ and optimal rates for the spline model using an approximation 1 n L L U where U is compact, n i=1 i i positive deﬁnite, self-adjoint and with dense inverse. See also [8, 28] that consider ill-posed inverse problems without noise using similar methods. In these papers the scaling regime for λ is given in n C− terms of the rate of decay of the eigenvalues of the inverse covariance (regularization) operator 1 (cid:107) · (cid:107) (cid:107)C− · (cid:107) (where = 1 ). 1 L2 There are many more recent results addressing the asymptotic properties of splines, including [9,17, 20, 24, 26, 33, 47, 48, 50, 51]. Many of these recent results concern the asymptotics of penalized splines where one ﬁxes the number of knot points as apposed to the smoothing spline case where the number of knots is equal to the number of data points. It is known that the special spline problem is equivalent to a white noise problem [7]. Strong con- vergence and rates for the white noise problem have been well studied see, for example, [2, 4, 16] and references therein. An interesting related result, due to Silverman [34], gives the convergence of the smoothing kernel. { } That is, we can write the estimator µn of µ given d(cid:88)ata (t , y ) n in the form i i i=1 n 1 µn(s) = K (s, t )y n i i n i=1 · for a Kernel K (see Lemma 2.8). Silverman showed that K ( , t) converges to some K uniformly − n n on [(cid:15), 1 (cid:15)] for every (cid:15) > 0 and each t (the result is valid for the special spline model and penalising the second derivative). Whilst this result gives intuition into how the kernel behaves it does not imply { } the convergence of the smoothing spline. Indeed, the convergence is not valid at the end points 0, 1 · → · and does not account for randomness in the observations y . In other words K ( , t) K( , t) does i n not imply the convergence of µn (or any characterisation of the limit such as we give in this paper as a solution to a variational problem). Silverman’s result is, however, valid for a larger range of λ than − we have here. For convergence of the kernel it is enough that 1 = o(n2 δ) for any δ > 0. Our results λ concerning the pointwise convergence of the smoothing spline hold for λ satisfying λ1 =→O(n12 ). One advantage of our approach is that we gain intuition in what happens when λ 0 too quickly. n Our results show a critical rate, with respect to the scaling of λ , at which the methodology is ill-posed n below this rate and well-posed at or above this rate. The second advantage of our approach is that, by using the Γ-convergence framework, as long as we can show that minimizers are uniformly bounded the convergence follows easily (we also need to show the Γ-limit is unique, but for our problem this is not − † difﬁcult). This is easier than showing, directly, that µn µ converges to zero. We are consequently able to employ simpler assumptions than those required by more direct arguments. The outline of this paper is as follows. In the next section we introduce some preliminary material. This starts by deﬁning the notation we use in the remainder of the paper. We then remind the reader of Gaˆteaux derivatives, the Γ-convergence framework and the spline methodology respectively. Section 3 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: particular

text: for

text: scaling

text: outside

text: this

text: regime

text: we

text: construct

text: (uniformly

text: bounded)

text: observation

text: operators

text: L

text: E

text: i

text: such

text: that

text: µn

text: Hence

text: without

text: further

text: assumptions

text: our

text: results

text: are

text: sharp.

text: We

text: note

text: that

text: these

text: results

text: have

text: practical

text: implications.

text: If

text: we

text: are

text: interested

text: in

text: estimating

text: µ

text: at

text: a

text: point

text: H∗

text: t

text: then

text: we

text: let

text: F

text: (µ)

text: µ(t)

text: where

text: F

text: In

text: this

text: setting

text: weak

text: convergence,

text: or

text: the

text: pointwise

text: form

text: considered

text: in

text: this

text: paper,

text: are

text: the

text: natural

text: modes

text: of

text: convergence

text: to

text: consider.

text: Whereas,

text: if

text: one

text: is

text: interested

text: in

text: a

text: global

text: approximation

text: of

text: µ

text: then

text: convergence

text: of

text: µn

text: µ

text: in

text: an

text: appropriate

text: norm

text: is

text: the

text: more

text: relevant.

text: The

text: two

text: formulations

text: imply

text: different

text: scaling

text: results

text: for

text: λ

text: n

text: There

text: are

text: many

text: results

text: in

text: the

text: ill-posed

text: inverse

text: problems

text: literature

text: that

text: may

text: be

text: applied

text: to

text: the

text: strong

text: convergence

text: of

text: the

text: general

text: spline

text: problem,

text: for

text: brevity

text: we

text: only

text: mention

text: those

text: most

text: relevant

text: to

text: this

text: work.

text: In

text: two

text: different

text: methods

text: of

text: estimating

text: λ

text: were

text: compared

text: as

text: n

text: using

text: the

text: general

text: spline

text: n

text: formulation.

text: The

text: reproducing

text: kernel

text: Hilbert

text: space

text: setting

text: was

text: used

text: in

text: which

text: also

text: discussed

text: the

text: probabilistic

text: interpretation

text: behind

text: the

text: estimator

text: µn.

text: In

text: the

text: authors

text: prove

text: the

text: strong

text: convergence

text: and

text: optimal

text: rates

text: for

text: the

text: spline

text: model

text: using

text: an

text: approximation

text: n

text: L

text: L

text: U

text: where

text: U

text: is

text: compact,

text: n

text: i

text: i

text: positive

text: deﬁnite,

text: self-adjoint

text: and

text: with

text: dense

text: inverse.

text: See

text: also

text: that

text: consider

text: ill-posed

text: inverse

text: problems

text: without

text: noise

text: using

text: similar

text: methods.

text: In

text: these

text: papers

text: the

text: scaling

text: regime

text: for

text: λ

text: is

text: given

text: in

text: n

text: C−

text: terms

text: of

text: the

text: rate

text: of

text: decay

text: of

text: the

text: eigenvalues

text: of

text: the

text: inverse

text: covariance

text: (regularization)

text: operator

text: (where

text: There

text: are

text: many

text: more

text: recent

text: results

text: addressing

text: the

text: asymptotic

text: properties

text: of

text: splines,

text: including

text: Many

text: of

text: these

text: recent

text: results

text: concern

text: the

text: asymptotics

text: of

text: penalized

text: splines

text: where

text: one

text: ﬁxes

text: the

text: number

text: of

text: knot

text: points

text: as

text: apposed

text: to

text: the

text: smoothing

text: spline

text: case

text: where

text: the

text: number

text: of

text: knots

text: is

text: equal

text: to

text: the

text: number

text: of

text: data

text: points.

text: It

text: is

text: known

text: that

text: the

text: special

text: spline

text: problem

text: is

text: equivalent

text: to

text: a

text: white

text: noise

text: problem

text: Strong

text: con-

text: vergence

text: and

text: rates

text: for

text: the

text: white

text: noise

text: problem

text: have

text: been

text: well

text: studied

text: see,

text: for

text: example,

text: and

text: references

text: therein.

text: An

text: interesting

text: related

text: result,

text: due

text: to

text: Silverman

text: gives

text: the

text: convergence

text: of

text: the

text: smoothing

text: kernel.

text: That

text: is,

text: we

text: can

text: write

text: the

text: estimator

text: µn

text: of

text: µ

text: given

text: (t

text: y

text: n

text: in

text: the

text: form

text: i

text: i

text: n

text: µn(s)

text: K

text: (s,

text: t

text: )y

text: n

text: i

text: i

text: n

text: for

text: a

text: Kernel

text: K

text: (see

text: Lemma

text: Silverman

text: showed

text: that

text: K

text: t)

text: converges

text: to

text: some

text: K

text: uniformly

text: n

text: n

text: on

text: for

text: every

text: and

text: each

text: t

text: (the

text: result

text: is

text: valid

text: for

text: the

text: special

text: spline

text: model

text: and

text: penalising

text: the

text: second

text: derivative).

text: Whilst

text: this

text: result

text: gives

text: intuition

text: into

text: how

text: the

text: kernel

text: behaves

text: it

text: does

text: not

text: imply

text: the

text: convergence

text: of

text: the

text: smoothing

text: spline.

text: Indeed,

text: the

text: convergence

text: is

text: not

text: valid

text: at

text: the

text: end

text: points

text: and

text: does

text: not

text: account

text: for

text: randomness

text: in

text: the

text: observations

text: y

text: In

text: other

text: words

text: K

text: t)

text: K(

text: t)

text: does

text: i

text: n

text: not

text: imply

text: the

text: convergence

text: of

text: µn

text: (or

text: any

text: characterisation

text: of

text: the

text: limit

text: such

text: as

text: we

text: give

text: in

text: this

text: paper

text: as

text: a

text: solution

text: to

text: a

text: variational

text: problem).

text: Silverman’s

text: result

text: is,

text: however,

text: valid

text: for

text: a

text: larger

text: range

text: of

text: λ

text: than

text: we

text: have

text: here.

text: For

text: convergence

text: of

text: the

text: kernel

text: it

text: is

text: enough

text: that

text: δ)

text: for

text: any

text: δ

text: Our

text: results

text: λ

text: concerning

text: the

text: pointwise

text: convergence

text: of

text: the

text: smoothing

text: spline

text: hold

text: for

text: λ

text: satisfying

text: λ1

text: One

text: advantage

text: of

text: our

text: approach

text: is

text: that

text: we

text: gain

text: intuition

text: in

text: what

text: happens

text: when

text: λ

text: too

text: quickly.

text: n

text: Our

text: results

text: show

text: a

text: critical

text: rate,

text: with

text: respect

text: to

text: the

text: scaling

text: of

text: λ

text: at

text: which

text: the

text: methodology

text: is

text: ill-posed

text: n

text: below

text: this

text: rate

text: and

text: well-posed

text: at

text: or

text: above

text: this

text: rate.

text: The

text: second

text: advantage

text: of

text: our

text: approach

text: is

text: that,

text: by

text: using

text: the

text: Γ-convergence

text: framework,

text: as

text: long

text: as

text: we

text: can

text: show

text: that

text: minimizers

text: are

text: uniformly

text: bounded

text: the

text: convergence

text: follows

text: easily

text: (we

text: also

text: need

text: to

text: show

text: the

text: Γ-limit

text: is

text: unique,

text: but

text: for

text: our

text: problem

text: this

text: is

text: not

text: difﬁcult).

text: This

text: is

text: easier

text: than

text: showing,

text: directly,

text: that

text: µn

text: µ

text: converges

text: to

text: zero.

text: We

text: are

text: consequently

text: able

text: to

text: employ

text: simpler

text: assumptions

text: than

text: those

text: required

text: by

text: more

text: direct

text: arguments.

text: The

text: outline

text: of

text: this

text: paper

text: is

text: as

text: follows.

text: In

text: the

text: next

text: section

text: we

text: introduce

text: some

text: preliminary

text: material.

text: This

text: starts

text: by

text: deﬁning

text: the

text: notation

text: we

text: use

text: in

text: the

text: remainder

text: of

text: the

text: paper.

text: We

text: then

text: remind

text: the

text: reader

text: of

text: Gaˆteaux

text: derivatives,

text: the

text: Γ-convergence

text: framework

text: and

text: the

text: spline

text: methodology

text: respectively.

text: Section

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [36] P. L. Speckman and D. Sun. Asymptotic properties of smoothing parameter selection in spline smoothing. Technical report, Department of Statistics, University of Missouri, 2001. [37] C. J. Stone. Optimal global rates of convergence for nonparametric regression. The Annals of Statistics, 10(4):1040–1053, 1982. [38] M. Thorpe and A. M. Johansen. Convergence and rates for ﬁxed-interval multiple-track smoothing using k-means type optimization. Electronic Journal of Statistics, 10(2):3693–3722, 2016. [39] M. Thorpe, F. Theil, A. M. Johansen, and N. Cade. Convergence of the k-means minimization problem using Γ-convergence. SIAM Journal on Applied Mathematics, 75(6):2444–2474, 2015. [40] F. I. Utreras. Optimal smoothing of noisy data using spline functions. SIAM Journal on Scientiﬁc and Statistical Computing, 2(3):349–362, 1981. [41] F. I. Utreras. Natural spline functions, their associated eigenvalue problem. Numerische Mathe- matik, 42(1):107–117, 1983. [42] F. I. Utreras. Smoothing noisy data under monotonicity constraints existence, characterization and convergence rates. Numerische Mathematik, 47(4):611–625, 1985. [43] G. Wahba. A comparison of GCV and GML for choosing the smoothing parameter in the general- ized spline smoothing problem. The Annals of Statistics, 13(4):1378–1402, 1985. [44] G. Wahba. Spline models for observational data. Society for Industrial and Applied Mathematics (SIAM), 1990. [45] G. Wahba and S. Wold. A completely automatic French curve: Fitting spline functions by cross- validation. Communications in Statistics, 4(1), 1975. [46] M. P. Wand. On the optimal amount of smoothing in penalised spline regression. Biometrika, 86(4):936–940, 1999. [47] X. Wang, J. Shen, and D. Ruppert. On the asymptotics of penalized spline smoothing. Electronic Journal of Statistics, 5:1–17, 2011. [48] L. Xiao, Y. Li, T. V. Apanasovich, and D. Ruppert. Local asymptotics of P-splines. arXiv preprint arXiv:1201.0708, 2012. [49] L. Xiao, Y. Li, and D. Ruppert. Fast bivariate P-splines: the sandwich smoother. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(3):577–599, 2013. [50] T. Yoshida and K. Naito. Asymptotics for penalized additive B-spline regression. Journal of the Japan Statistical Society, 42(1):81–107, 2012. [51] T. Yoshida and K. Naito. Asymptotics for penalised splines in generalised additive models. Journal of Nonparametric Statistics, 26(2):269–289, 2014. 23

text: P.

text: L.

text: Speckman

text: and

text: D.

text: Sun.

text: Asymptotic

text: properties

text: of

text: smoothing

text: parameter

text: selection

text: in

text: spline

text: smoothing.

text: Technical

text: report,

text: Department

text: of

text: Statistics,

text: University

text: of

text: Missouri,

text: C.

text: J.

text: Stone.

text: Optimal

text: global

text: rates

text: of

text: convergence

text: for

text: nonparametric

text: regression.

text: The

text: Annals

text: of

text: Statistics,

text: M.

text: Thorpe

text: and

text: A.

text: M.

text: Johansen.

text: Convergence

text: and

text: rates

text: for

text: ﬁxed-interval

text: multiple-track

text: smoothing

text: using

text: k-means

text: type

text: optimization.

text: Electronic

text: Journal

text: of

text: Statistics,

text: M.

text: Thorpe,

text: F.

text: Theil,

text: A.

text: M.

text: Johansen,

text: and

text: N.

text: Cade.

text: Convergence

text: of

text: the

text: k-means

text: minimization

text: problem

text: using

text: Γ-convergence.

text: SIAM

text: Journal

text: on

text: Applied

text: Mathematics,

text: F.

text: I.

text: Utreras.

text: Optimal

text: smoothing

text: of

text: noisy

text: data

text: using

text: spline

text: functions.

text: SIAM

text: Journal

text: on

text: Scientiﬁc

text: and

text: Statistical

text: Computing,

text: F.

text: I.

text: Utreras.

text: Natural

text: spline

text: functions,

text: their

text: associated

text: eigenvalue

text: problem.

text: Numerische

text: Mathe-

text: matik,

text: F.

text: I.

text: Utreras.

text: Smoothing

text: noisy

text: data

text: under

text: monotonicity

text: constraints

text: existence,

text: characterization

text: and

text: convergence

text: rates.

text: Numerische

text: Mathematik,

text: G.

text: Wahba.

text: A

text: comparison

text: of

text: GCV

text: and

text: GML

text: for

text: choosing

text: the

text: smoothing

text: parameter

text: in

text: the

text: general-

text: ized

text: spline

text: smoothing

text: problem.

text: The

text: Annals

text: of

text: Statistics,

text: G.

text: Wahba.

text: Spline

text: models

text: for

text: observational

text: data.

text: Society

text: for

text: Industrial

text: and

text: Applied

text: Mathematics

text: (SIAM),

text: G.

text: Wahba

text: and

text: S.

text: Wold.

text: A

text: completely

text: automatic

text: French

text: curve:

text: Fitting

text: spline

text: functions

text: by

text: cross-

text: validation.

text: Communications

text: in

text: Statistics,

text: M.

text: P.

text: Wand.

text: On

text: the

text: optimal

text: amount

text: of

text: smoothing

text: in

text: penalised

text: spline

text: regression.

text: Biometrika,

text: X.

text: Wang,

text: J.

text: Shen,

text: and

text: D.

text: Ruppert.

text: On

text: the

text: asymptotics

text: of

text: penalized

text: spline

text: smoothing.

text: Electronic

text: Journal

text: of

text: Statistics,

text: L.

text: Xiao,

text: Y.

text: Li,

text: T.

text: V.

text: Apanasovich,

text: and

text: D.

text: Ruppert.

text: Local

text: asymptotics

text: of

text: P-splines.

text: arXiv

text: preprint

text: L.

text: Xiao,

text: Y.

text: Li,

text: and

text: D.

text: Ruppert.

text: Fast

text: bivariate

text: P-splines:

text: the

text: sandwich

text: smoother.

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society:

text: Series

text: B

text: (Statistical

text: Methodology),

text: T.

text: Yoshida

text: and

text: K.

text: Naito.

text: Asymptotics

text: for

text: penalized

text: additive

text: B-spline

text: regression.

text: Journal

text: of

text: the

text: Japan

text: Statistical

text: Society,

text: T.

text: Yoshida

text: and

text: K.

text: Naito.

text: Asymptotics

text: for

text: penalised

text: splines

text: in

text: generalised

text: additive

text: models.

text: Journal

text: of

text: Nonparametric

text: Statistics,


text: ASYMPTOTIC BEHAVIOR OF INTERFACE SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS WITH NONLINEAR FORCING TERM1S1 (cid:3) and the proof is completed. Let us stress that, in the proof of Theorem 3.2, the negativity of λε is crucial in the 1 construction of a stable family of generators. We also take advantage of the expression of Q ε, where the ﬁrst order space derivative of v does not appear; indeed, this allows us to R ∞ estimate the nonlinear term ε via the L - norm of v. If we start from an initial datum v with a weaker regularity, precisely belonging to 0 L2, we can prove an estimate analogous to (3.8) for the L2 norm of the solution, but the following additional technical hypothesis is needed. ∈ · · H4.1 Given ξ I, let ϕε( ; ξ) and ψε( ; ξ) be a sequence of eigenfunction for the operators Lε and Lε,∗ resp(cid:88)ectkively; we assukm(cid:88)e ξ ξ (cid:104) (cid:105) (cid:104) (cid:105) ≤ (3.9) ∂ ψε, ϕε 2 = ψε, ∂ ϕε 2 C. ξ k j k ξ j j j for all k and for some constant C independent on the parameter ξ. Theorem 3.3. Let the couple (ξ, v) be the solution to initial-value problem (3.1). If the hypotheses H1-2-3-4.1 are satis(cid:18)ﬁed, then, for e(cid:18)ve(cid:90)ry ε suﬃcientl(cid:19)y small (cid:19)there exists a time ∈ ≤ T ε such that, for every v L2(I) and for every t T ε, there holds for the solution v 0 | − | ≤ | | t | | v z L2(t) C(cid:88) Ωε L∞ + exp 0 λε1(ξ(τ)) dτ v0 2L2 , where the function z is deﬁned as z(x, t) := v (0) exp (λε(ξ(τ)) dτ) ϕε(x; ξ(t)). k k k ≥ k 2 | |− ∞ → Moreover, the time T ε is of order sup λε(ξ) 1, hence diverging to + as ε 0. ∈ 1 ξ I (cid:88) Proof of Theorem 3.3. Setting v(x, t) = v (t) ϕε(x, ξ(t)), j j j we obtain an inﬁnite-dimensional diﬀerential system for the coeﬃcients v j dv (cid:104) (cid:105) (cid:104) (cid:105) (3.10) k = λε(ξ) v + ψε, F + ψε, F dt k k k 1 k 2 (cid:88) (cid:110) (cid:111) (cid:88)(cid:16) (cid:88) (cid:17) where, omitting the dependencies for shortness, F1 := Hε +(cid:88)vj Mεξ ϕεj − ∂ξϕεj d(cid:26)dξt = Hε − θε aj + bj(cid:96) v(cid:96)(cid:27) vj, j  j (cid:96) (cid:104) Q (cid:105) (cid:104) (cid:105) Q − ψε, ε − ∂ ψε, v 2 Q − N F := ε ∂ ϕεv + ∂ Uε − (cid:104)1 (cid:105) θε −ξ(cid:104) 1 (cid:105) = ε ε. 2 ξ j j ξ 1 ∂ ψε, v 1 ∂ ψε, v j ξ 1 ξ 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ASYMPTOTIC

text: BEHAVIOR

text: OF

text: INTERFACE

text: SOLUTIONS

text: TO

text: QUASILINEAR

text: PARABOLIC

text: EQUATIONS

text: WITH

text: NONLINEAR

text: FORCING

text: and

text: the

text: proof

text: is

text: completed.

text: Let

text: us

text: stress

text: that,

text: in

text: the

text: proof

text: of

text: Theorem

text: the

text: negativity

text: of

text: λε

text: is

text: crucial

text: in

text: the

text: construction

text: of

text: a

text: stable

text: family

text: of

text: generators.

text: We

text: also

text: take

text: advantage

text: of

text: the

text: expression

text: of

text: Q

text: ε,

text: where

text: the

text: ﬁrst

text: order

text: space

text: derivative

text: of

text: v

text: does

text: not

text: appear;

text: indeed,

text: this

text: allows

text: us

text: to

text: R

text: estimate

text: the

text: nonlinear

text: term

text: ε

text: via

text: the

text: L

text: norm

text: of

text: v.

text: If

text: we

text: start

text: from

text: an

text: initial

text: datum

text: v

text: with

text: a

text: weaker

text: regularity,

text: precisely

text: belonging

text: to

text: we

text: can

text: prove

text: an

text: estimate

text: analogous

text: to

text: for

text: the

text: norm

text: of

text: the

text: solution,

text: but

text: the

text: following

text: additional

text: technical

text: hypothesis

text: is

text: needed.

text: Given

text: ξ

text: I,

text: let

text: ϕε(

text: ξ)

text: and

text: ψε(

text: ξ)

text: be

text: a

text: sequence

text: of

text: eigenfunction

text: for

text: the

text: operators

text: Lε

text: and

text: Lε,∗

text: we

text: ξ

text: ξ

text: ψε,

text: ϕε

text: ψε,

text: ϕε

text: C.

text: ξ

text: k

text: j

text: k

text: ξ

text: j

text: j

text: j

text: for

text: all

text: k

text: and

text: for

text: some

text: constant

text: C

text: independent

text: on

text: the

text: parameter

text: ξ.

text: Theorem

text: Let

text: the

text: couple

text: (ξ,

text: v)

text: be

text: the

text: solution

text: to

text: initial-value

text: problem

text: If

text: the

text: hypotheses

text: are

text: then,

text: for

text: ε

text: small

text: exists

text: a

text: time

text: T

text: ε

text: such

text: that,

text: for

text: every

text: v

text: and

text: for

text: every

text: t

text: T

text: ε,

text: there

text: holds

text: for

text: the

text: solution

text: v

text: t

text: v

text: z

text: Ωε

text: L∞

text: exp

text: dτ

text: where

text: the

text: function

text: z

text: is

text: deﬁned

text: as

text: z(x,

text: t)

text: v

text: exp

text: (λε(ξ(τ))

text: dτ)

text: ϕε(x;

text: ξ(t)).

text: k

text: k

text: k

text: k

text: Moreover,

text: the

text: time

text: T

text: ε

text: is

text: of

text: order

text: sup

text: λε(ξ)

text: hence

text: diverging

text: to

text: as

text: ε

text: ξ

text: I

text: Proof

text: of

text: Theorem

text: Setting

text: v(x,

text: t)

text: v

text: (t)

text: ϕε(x,

text: ξ(t)),

text: j

text: j

text: j

text: we

text: obtain

text: an

text: inﬁnite-dimensional

text: diﬀerential

text: system

text: for

text: the

text: coeﬃcients

text: v

text: j

text: dv

text: k

text: λε(ξ)

text: v

text: ψε,

text: F

text: ψε,

text: F

text: dt

text: k

text: k

text: k

text: k

text: where,

text: omitting

text: the

text: dependencies

text: for

text: shortness,

text: Hε

text: Mεξ

text: ϕεj

text: ∂ξϕεj

text: Hε

text: θε

text: aj

text: vj,

text: j

text: j

text: Q

text: Q

text: ψε,

text: ε

text: ψε,

text: v

text: Q

text: N

text: F

text: ε

text: ϕεv

text: Uε

text: θε

text: ε

text: ε.

text: ξ

text: j

text: j

text: ξ

text: ψε,

text: v

text: ψε,

text: v

text: j

text: ξ

text: ξ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ASYMPTOTIC BEHAVIOR OF INTERFACE SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS WITH NONLINEAR FORCING TERMS3 Motivated by this, we expect that also in the more general setting a(x) satisfying (1.2), all the discontinuities that appear at the hyperbolic level ε = 0 shall eventually turn out into smooth internal layers, and that a metastable behavior will be observable in the vanishing viscosity limit. In order to characterize the slow dynamics of solutions to (1.1), we mean to adapt the theory developed in [15] for parabolic equations to our more general setting; this strategy dates back the work of J. Carr and R. L. Pego [6] and can be summarized as follows. { } The principal idea is to construct a family Uε(x, ξ ) ∈ , i = 1, ....., N, of so called i ξi I approximate steady states for the problem, and to linearize the original equation around an element of this family. With approximate steady state for (1.1), we refer to a solution that solves the stationary equation up to an error that is small in a sense that will be speciﬁed later. The parameters ξ represent the location of the interfaces. i The aim of this construction is to separate the two distinct phases of the dynamics. Firstly, we mean to understand what happens far from the steady state solution, when the interfaces are formed; subsequently, we want to follow the evolution of the layered solution towards the asymptotic limit. In particular, we describe such slow motion by obtaining an equation for the positions of the interfaces ξ (t), i = 1, ....., N. For the sake of simplicity, i in this paper we restrict our analysis to the case of a solution with a single shock layer ∈ located in ξ I, the general case being similar (see, for example, [6]). { } After the family Uε is given, a study of the eigenvalue problem associated to the L linearized operator ε obtained from the linearization is needed. Indeed, we want to describe the dynamics of solutions located far from the equilibrium conﬁguration of the L system and, by performing a spectral analysis of ε, we are able to show that the speed rate of convergence of the solutions towards the asymptotic conﬁguration is small in ε. We close this Introduction with an overview of the paper. In Section 2 we present the general strategy we develope in order to describe the long time behavior of solutions { } belonging to a neighborhood of a family of approximate steady states Uε(x; ξ) ∈ . By ξ I linearizing the original equation (1.1) around an element of the family, i.e. by looking for a solution u on the form u = Uε + v, and by using an adapted version of the projection method in order to remove the growing components of the perturbation v, we obtain a cou- pled system for the variables (ξ, v), whose analysis is performed in the subsequent section. In particular, in Section 3, we state and prove Theorems 3.2, 3.3, 3.4 and 3.7, providing diﬀerent estimates for the perturbation v, depending on the choice of the nonlinear term G(u, ∂ u) and on the sign of the ﬁrst eigenvalue of the linearized operator obtained from x the linearization. Speciﬁcally, since we are taking into account also the nonlinear terms in v (arising from the linearization), we will show that both the form of G and the sign of λε inﬂuence the speed rate of convergence to zero of the perturbation. The estimates 1 on v will be used to decouple the nonlinear system for the variables (ξ; v): we end up with a one-dimensional equation of motion for the variable ξ, whose analysis is addressed in Proposition 3.11. In particular, the metastable behavior of the solution is described through the convergence of the interface location towards its equilibrium conﬁguration. Again, the speed rate of this motion is inﬂuenced by the explicit form of G and by the sign of λε. 1 Precisely, our results can be summarized in the following Theorem.

text: ASYMPTOTIC

text: BEHAVIOR

text: OF

text: INTERFACE

text: SOLUTIONS

text: TO

text: QUASILINEAR

text: PARABOLIC

text: EQUATIONS

text: WITH

text: NONLINEAR

text: FORCING

text: Motivated

text: by

text: this,

text: we

text: expect

text: that

text: also

text: in

text: the

text: more

text: general

text: setting

text: a(x)

text: satisfying

text: all

text: the

text: discontinuities

text: that

text: appear

text: at

text: the

text: hyperbolic

text: level

text: ε

text: shall

text: eventually

text: turn

text: out

text: into

text: smooth

text: internal

text: layers,

text: and

text: that

text: a

text: metastable

text: behavior

text: will

text: be

text: observable

text: in

text: the

text: vanishing

text: viscosity

text: limit.

text: In

text: order

text: to

text: characterize

text: the

text: slow

text: dynamics

text: of

text: solutions

text: to

text: we

text: mean

text: to

text: adapt

text: the

text: theory

text: developed

text: in

text: for

text: parabolic

text: equations

text: to

text: our

text: more

text: general

text: setting;

text: this

text: strategy

text: dates

text: back

text: the

text: work

text: of

text: J.

text: Carr

text: and

text: R.

text: L.

text: Pego

text: and

text: can

text: be

text: summarized

text: as

text: follows.

text: The

text: principal

text: idea

text: is

text: to

text: construct

text: a

text: family

text: Uε(x,

text: ξ

text: i

text: N,

text: of

text: so

text: called

text: i

text: ξi

text: I

text: approximate

text: steady

text: states

text: for

text: the

text: problem,

text: and

text: to

text: linearize

text: the

text: original

text: equation

text: around

text: an

text: element

text: of

text: this

text: family.

text: With

text: approximate

text: steady

text: state

text: for

text: we

text: refer

text: to

text: a

text: solution

text: that

text: solves

text: the

text: stationary

text: equation

text: up

text: to

text: an

text: error

text: that

text: is

text: small

text: in

text: a

text: sense

text: that

text: will

text: be

text: speciﬁed

text: later.

text: The

text: parameters

text: ξ

text: represent

text: the

text: location

text: of

text: the

text: interfaces.

text: i

text: The

text: aim

text: of

text: this

text: construction

text: is

text: to

text: separate

text: the

text: two

text: distinct

text: phases

text: of

text: the

text: dynamics.

text: Firstly,

text: we

text: mean

text: to

text: understand

text: what

text: happens

text: far

text: from

text: the

text: steady

text: state

text: solution,

text: when

text: the

text: interfaces

text: are

text: formed;

text: subsequently,

text: we

text: want

text: to

text: follow

text: the

text: evolution

text: of

text: the

text: layered

text: solution

text: towards

text: the

text: asymptotic

text: limit.

text: In

text: particular,

text: we

text: describe

text: such

text: slow

text: motion

text: by

text: obtaining

text: an

text: equation

text: for

text: the

text: positions

text: of

text: the

text: interfaces

text: ξ

text: (t),

text: i

text: N.

text: For

text: the

text: sake

text: of

text: simplicity,

text: i

text: in

text: this

text: paper

text: we

text: restrict

text: our

text: analysis

text: to

text: the

text: case

text: of

text: a

text: solution

text: with

text: a

text: single

text: shock

text: layer

text: located

text: in

text: ξ

text: I,

text: the

text: general

text: case

text: being

text: similar

text: (see,

text: for

text: example,

text: After

text: the

text: family

text: Uε

text: is

text: given,

text: a

text: study

text: of

text: the

text: eigenvalue

text: problem

text: associated

text: to

text: the

text: L

text: linearized

text: operator

text: ε

text: obtained

text: from

text: the

text: linearization

text: is

text: needed.

text: Indeed,

text: we

text: want

text: to

text: describe

text: the

text: dynamics

text: of

text: solutions

text: located

text: far

text: from

text: the

text: equilibrium

text: conﬁguration

text: of

text: the

text: L

text: system

text: and,

text: by

text: performing

text: a

text: spectral

text: analysis

text: of

text: ε,

text: we

text: are

text: able

text: to

text: show

text: that

text: the

text: speed

text: rate

text: of

text: convergence

text: of

text: the

text: solutions

text: towards

text: the

text: asymptotic

text: conﬁguration

text: is

text: small

text: in

text: ε.

text: We

text: close

text: this

text: Introduction

text: with

text: an

text: overview

text: of

text: the

text: paper.

text: In

text: Section

text: we

text: present

text: the

text: general

text: strategy

text: we

text: develope

text: in

text: order

text: to

text: describe

text: the

text: long

text: time

text: behavior

text: of

text: solutions

text: belonging

text: to

text: a

text: neighborhood

text: of

text: a

text: family

text: of

text: approximate

text: steady

text: states

text: Uε(x;

text: ξ)

text: By

text: ξ

text: I

text: linearizing

text: the

text: original

text: equation

text: around

text: an

text: element

text: of

text: the

text: family,

text: i.e.

text: by

text: looking

text: for

text: a

text: solution

text: u

text: on

text: the

text: form

text: u

text: Uε

text: v,

text: and

text: by

text: using

text: an

text: adapted

text: version

text: of

text: the

text: projection

text: method

text: in

text: order

text: to

text: remove

text: the

text: growing

text: components

text: of

text: the

text: perturbation

text: v,

text: we

text: obtain

text: a

text: cou-

text: pled

text: system

text: for

text: the

text: variables

text: (ξ,

text: v),

text: whose

text: analysis

text: is

text: performed

text: in

text: the

text: subsequent

text: section.

text: In

text: particular,

text: in

text: Section

text: we

text: state

text: and

text: prove

text: Theorems

text: and

text: providing

text: diﬀerent

text: estimates

text: for

text: the

text: perturbation

text: v,

text: depending

text: on

text: the

text: choice

text: of

text: the

text: nonlinear

text: term

text: G(u,

text: u)

text: and

text: on

text: the

text: sign

text: of

text: the

text: ﬁrst

text: eigenvalue

text: of

text: the

text: linearized

text: operator

text: obtained

text: from

text: x

text: the

text: linearization.

text: Speciﬁcally,

text: since

text: we

text: are

text: taking

text: into

text: account

text: also

text: the

text: nonlinear

text: terms

text: in

text: v

text: (arising

text: from

text: the

text: linearization),

text: we

text: will

text: show

text: that

text: both

text: the

text: form

text: of

text: G

text: and

text: the

text: sign

text: of

text: λε

text: inﬂuence

text: the

text: speed

text: rate

text: of

text: convergence

text: to

text: zero

text: of

text: the

text: perturbation.

text: The

text: estimates

text: on

text: v

text: will

text: be

text: used

text: to

text: decouple

text: the

text: nonlinear

text: system

text: for

text: the

text: variables

text: (ξ;

text: v):

text: we

text: end

text: up

text: with

text: a

text: one-dimensional

text: equation

text: of

text: motion

text: for

text: the

text: variable

text: ξ,

text: whose

text: analysis

text: is

text: addressed

text: in

text: Proposition

text: In

text: particular,

text: the

text: metastable

text: behavior

text: of

text: the

text: solution

text: is

text: described

text: through

text: the

text: convergence

text: of

text: the

text: interface

text: location

text: towards

text: its

text: equilibrium

text: conﬁguration.

text: Again,

text: the

text: speed

text: rate

text: of

text: this

text: motion

text: is

text: inﬂuenced

text: by

text: the

text: explicit

text: form

text: of

text: G

text: and

text: by

text: the

text: sign

text: of

text: λε.

text: Precisely,

text: our

text: results

text: can

text: be

text: summarized

text: in

text: the

text: following

text: Theorem.


text: A28SYMPTOTIC BEHAVIOR OF INTERFACE SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS WITH NONLINEAR FORCING TERMS (cid:90) (cid:18) (cid:20)(cid:90) (cid:90) (cid:21)(cid:19) ≤ where we used a(x) β. Moreover there holds (cid:48) − (cid:48) (cid:90) (cid:96) 1 ξ f (u ) x f (u+) I0 = exp dy + dy dx − ε − a(y) a(y) (cid:96) (cid:96) ξ ≥ (cid:96) (cid:48) − (cid:90) (cid:48) − ef (u )(ξ+(cid:96))/βε +f (u+)(x ξ)/βε dx − (cid:96) (cid:48) − (cid:96) (cid:48) − ∼ (cid:48) − 1 f (u )(ξ+(cid:96)) 1 f (u+)(x ξ) 1 f (u )(ξ+(cid:96)) = eβ ε eβ ε dx eβ ε ε. − (cid:96) | (cid:48) − (cid:48) | ≤ In particular, if we suppose f (Uε) f (U0) c ε for some c > 0, we end up with L1 √0 0 | |− | | ≥ (cid:0) (cid:1) Kε L21 ψ0ε L2 C1 ε eC2/ε. N Thus, for th√e ﬁrst eigenvalue µε of the self-adjoint operator ε there holds the estimate | | ≤ − 1 ξ µε1 C1 Lε eC2/ε 1 for some−posNitive constant C1, C2. As a consequence, since the spectrum σ( ε) coincides with ε 1σ( ε), there holds ξ ξ − − ≤ (4.12) C e C/ε λε < 0. 1 (cid:48) | (cid:48) − (cid:48) | ≤ Remark 4.4. Since f is a continuous function, the request f (Uε) f (U0) c ε is | − | ≤ L1 0 satisﬁed if we require Uε U0 c ε, which is consistent with the convergence of Uε → L1 0 to U0 as ε 0. We mean to give an estimate on the behavior of the second and subsequent eigenvalues L of the operator ε. To this aim, we need some additional assumptions on the limiting ξ (cid:48) → behavior of the functions aε(x) =f (Uε(x)) as ε 0+. Precisely, inspired by [15], we ∈ − ∈ ∞ − suppose that aε C1([ (cid:96), (cid:96)]) and a L ([ (cid:96), (cid:96)]) satisfy the following hypotheses:  ∈ − \ { } ∈ − \ { }  i. aε C2([ (cid:96), (cid:96)] ξ ), a C1([ (cid:96), (cid:96)] ξ ) and  daε d2aε da −  , < 0 < aε, in ( (cid:96), ξ)  dx dx2 dx daε da d2aε aε, , < 0 < in (ξ, (cid:96)), (cid:12) (cid:12) dx dx dx2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ii. there exist the left(cid:12)/righ(cid:12)t ﬁrst order derivatives of a(cid:12) ε at ξ an(cid:12)d (cid:12) (cid:12) (cid:12) (cid:12) daε ≤ daε ± ε C and lim inf ε (ξ ) > 0. → dx ε 0+ dx | − | ≥ iii. for any C > 0 there exists c > 0 such that, if x ξ c ε, then 0 0 | (cid:48) − (cid:48) | ≤ f (Uε) f (U0) C ε, (cid:48) (cid:48) (cid:48) where we recall f (U0)(x) := f (u−)χ − + f (u )χ . ( (cid:96),ξ) + (ξ,(cid:96)) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BEHAVIOR

text: OF

text: INTERFACE

text: SOLUTIONS

text: TO

text: QUASILINEAR

text: PARABOLIC

text: EQUATIONS

text: WITH

text: NONLINEAR

text: FORCING

text: TERMS

text: where

text: we

text: used

text: a(x)

text: β.

text: Moreover

text: there

text: holds

text: ξ

text: f

text: (u

text: x

text: f

text: (u+)

text: exp

text: dy

text: dy

text: dx

text: ε

text: a(y)

text: a(y)

text: ξ

text: ef

text: (u

text: +f

text: (u+)(x

text: ξ)/βε

text: dx

text: f

text: (u

text: f

text: (u+)(x

text: ξ)

text: f

text: (u

text: eβ

text: ε

text: eβ

text: ε

text: dx

text: eβ

text: ε

text: ε.

text: In

text: particular,

text: if

text: we

text: suppose

text: f

text: (Uε)

text: f

text: c

text: ε

text: for

text: some

text: c

text: we

text: end

text: up

text: with

text: √0

text: Kε

text: ψ0ε

text: ε

text: N

text: Thus,

text: for

text: th√e

text: ﬁrst

text: eigenvalue

text: µε

text: of

text: the

text: self-adjoint

text: operator

text: ε

text: there

text: holds

text: the

text: estimate

text: ξ

text: µε1

text: Lε

text: for

text: some−posNitive

text: constant

text: As

text: a

text: consequence,

text: since

text: the

text: spectrum

text: σ(

text: ε)

text: coincides

text: with

text: ε

text: ε),

text: there

text: holds

text: ξ

text: ξ

text: C

text: e

text: C/ε

text: λε

text: Remark

text: Since

text: f

text: is

text: a

text: continuous

text: function,

text: the

text: request

text: f

text: (Uε)

text: f

text: c

text: ε

text: is

text: satisﬁed

text: if

text: we

text: require

text: Uε

text: c

text: ε,

text: which

text: is

text: consistent

text: with

text: the

text: convergence

text: of

text: Uε

text: to

text: as

text: ε

text: We

text: mean

text: to

text: give

text: an

text: estimate

text: on

text: the

text: behavior

text: of

text: the

text: second

text: and

text: subsequent

text: eigenvalues

text: L

text: of

text: the

text: operator

text: ε.

text: To

text: this

text: aim,

text: we

text: need

text: some

text: additional

text: assumptions

text: on

text: the

text: limiting

text: ξ

text: behavior

text: of

text: the

text: functions

text: aε(x)

text: =f

text: (Uε(x))

text: as

text: ε

text: Precisely,

text: inspired

text: by

text: we

text: suppose

text: that

text: aε

text: and

text: a

text: L

text: satisfy

text: the

text: following

text: hypotheses:

text: i.

text: aε

text: ξ

text: a

text: ξ

text: and

text: daε

text: da

text: aε,

text: in

text: ξ)

text: dx

text: dx

text: daε

text: da

text: aε,

text: in

text: (ξ,

text: dx

text: dx

text: ii.

text: there

text: exist

text: the

text: ﬁrst

text: order

text: derivatives

text: of

text: ε

text: at

text: ξ

text: daε

text: daε

text: ε

text: C

text: and

text: lim

text: inf

text: ε

text: (ξ

text: dx

text: ε

text: dx

text: iii.

text: for

text: any

text: C

text: there

text: exists

text: c

text: such

text: that,

text: if

text: x

text: ξ

text: c

text: ε,

text: then

text: f

text: (Uε)

text: f

text: C

text: ε,

text: where

text: we

text: recall

text: f

text: f

text: (u−)χ

text: f

text: (u

text: )χ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A34SYMPTOTIC BEHAVIOR OF INTERFACE SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS WITH NONLINEAR FORCING TERMS [9] Kim, Y.J.; Tzavaras, A.E.; Diﬀusive N-waves and metastability in the Burgers equation, SIAM J. Math. Anal. 33 (2001) no. 3, 607–633. [10] Kreiss, G., Kreiss, H.-O.; Convergence to steady state of solutions of Burgers’ equation, Appl. Numer. Math. 2 (1986) no. 3-5, 161–179. [11] Kreiss, G., Kreiss, H.-O.; Lorenz, J.; Stability of viscous shocks on ﬁnite intervals, Arch. Ration. Mech. Anal. 187 (2008) no. 1, 157–183. [12] Kruzkov S. N.; First order quasilinear equations in several independent variables, Mat. Sbornik 81 (1970), 228–255. Math. USSR Sbornik 10 (1970), 217–243. [13] Laforgue J.G.L.; O’Malley R.E. Jr.; Shock layer movement for Burgers equation, Perturbations meth- ods in physical mathematics (Troy, NY, 1993). SIAM J. Appl. Math. 55 (1995) no. 2, 332–347. [14] Mascia, C.; Terracina, A.; Large-time behavior for conservation laws with source in a bounded domain, J. Diﬀerential Equations 159 (1999), no. 2, 485–514. [15] Mascia C.; Strani M.; Metastability for nonlinear parabolic equations with application to scalar con- servation laws , SIAM J. Math. Anal. 45 (2013), no. 5, 3084–3113. [16] Otto, F.; Reznikoﬀ, M.G.; Slow motion of gradient ﬂows, J. Diﬀerential Equations 237 (2006) no. 2, 372–420. [17] Pazy A. (1983). Semigroups of Linear Operators and Applications to Partial Diﬀerential Equations, Applied Mathematical Science 44, Springer-Verlag, New York. [18] Pego, R. L.; Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A 422 (1989) no. 1863, 261–278. [19] Reyna L.G.; Ward M.J.; On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48 (1995), no. 2, 79–120. [20] Risler E., Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 25 (2008), no. 2, 381–424. [21] Sternberg, P., Vector-valued local minimizers of nonconvex variational problems, Current directions in nonlinear partial diﬀerential equations (Provo, UT, 1987). Rocky Mountain J. Math. 21 (1991), no. 2, 799–807. [22] Strani, M.; Slow motion of internal shock layers for the Jin-Xin system in one space dymension, J. Dyn.Diﬀ. Eq., 27 (2015), no. 1, pp. 1-27. [23] Strani, M.; Nonlinear metastability for a parabolic system of reaction-diﬀusion equations , submitted 2014. [24] Strani M., Metastable dynamics of internal interfaces for a convection-reaction-diﬀusion equation, submitted 2014. [25] Sun, X.; Ward, M. J.; Metastability for a generalized Burgers equation with application to propagating ﬂame fronts, European J. Appl. Math. 10 (1999), no. 1, 27–53

text: BEHAVIOR

text: OF

text: INTERFACE

text: SOLUTIONS

text: TO

text: QUASILINEAR

text: PARABOLIC

text: EQUATIONS

text: WITH

text: NONLINEAR

text: FORCING

text: TERMS

text: Kim,

text: Y.J.;

text: Tzavaras,

text: A.E.;

text: Diﬀusive

text: N-waves

text: and

text: metastability

text: in

text: the

text: Burgers

text: equation,

text: SIAM

text: J.

text: Math.

text: Anal.

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text: ASYMPTOTIC BEHAVIOR OF INTERFACE SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS WITH NONLINEAR FORCING TERMS9 L H4. Concerning the solution z to the linear problem ∂ z = εz, we require that there ∈ t ξ exists νε > 0 such that, for all ξ I, there exist a constant C¯ such that | | ≤ | | − ∀ ∈ z(t) C¯ z e νεt, ξ I L2 0 L2 Remark 3.1. The constant C¯ could depend on ξ. In this speciﬁc case, since ξ belongs to (cid:55)→ a bounded interval of the real line, if we suppose that ξ C is a continuous function, ξ(t) then there exists a maximum of C in I, namely C¯. ξ | | ∞ Theorem 3.2. Let hypotheses H1-2-3-4 be satisﬁed and∞let v0 →L∞ < + . Then, for ε suﬃciently small, there exists a time T ε diverging to + as ε 0, such that, for all ≤ t T ε the solution v to (3.1) satisﬁes | | ≤ | | − | | v ∞(t) C Ωε ∞t + e µεt v0 ∞, L L L for some positive constant C and { } − | | µε := sup λε(ξ) C Ωε ∞ > 0. 1 L ξ The proof of Theorem 3.2 we present here is based on the theory of stable families of generators, ﬁrstly developed by Pazy in [17]; it is a generalization of the theory of semigroups for evolution systems of the form ∂ u = Lu, when the linear operator L t depends on time. We refer to Appendix A for the deﬁnitions of the tools we shall use in the following. M Proof of Theorem 3.2. First of all, let us notice that ε is a bounded operator that sat- ξ isﬁes the estimate (cid:107)M (cid:107) ≤ | | ≤ | | ∀ ∈ (3.3) εξ L(L2;R) c1 θε(ξ) c1 Ωε L∞, ξ I. and Hε(ξ) is such that | | ≤ | | (3.4) Hε ∞ c2 Ωε ∞, L L for some positive constants c and c . Moreover, concerning the nonlinear terms ρε and R 1 2Q ε and because of the speciﬁc form of ε, there follows |R | ≤ | | (3.5) ε ∞ C v 2∞. L L L M Next, we want to show that ε + ε is the inﬁnitesimal generator of a C semigroup T ξ ξ L 0 (t, s). To this aim, concerning the eigenvalues of the linear operator ε, we know that ξ → ∈ ξ λε(ξ) is negative and goes to zero as ε 0, for all ξ I. Hence, deﬁning Λε := sup λε(ξ), 1 1 ξ 1

text: ASYMPTOTIC

text: BEHAVIOR

text: OF

text: INTERFACE

text: SOLUTIONS

text: TO

text: QUASILINEAR

text: PARABOLIC

text: EQUATIONS

text: WITH

text: NONLINEAR

text: FORCING

text: L

text: Concerning

text: the

text: solution

text: z

text: to

text: the

text: linear

text: problem

text: z

text: εz,

text: we

text: require

text: that

text: there

text: t

text: ξ

text: exists

text: νε

text: such

text: that,

text: for

text: all

text: ξ

text: I,

text: there

text: exist

text: a

text: constant

text: C¯

text: such

text: that

text: z(t)

text: C¯

text: z

text: e

text: νεt,

text: ξ

text: I

text: Remark

text: The

text: constant

text: C¯

text: could

text: depend

text: on

text: ξ.

text: In

text: this

text: speciﬁc

text: case,

text: since

text: ξ

text: belongs

text: to

text: a

text: bounded

text: interval

text: of

text: the

text: real

text: line,

text: if

text: we

text: suppose

text: that

text: ξ

text: C

text: is

text: a

text: continuous

text: function,

text: ξ(t)

text: then

text: there

text: exists

text: a

text: maximum

text: of

text: C

text: in

text: I,

text: namely

text: C¯.

text: ξ

text: Theorem

text: Let

text: hypotheses

text: be

text: satisﬁed

text: and∞let

text: →L∞

text: Then,

text: for

text: ε

text: suﬃciently

text: small,

text: there

text: exists

text: a

text: time

text: T

text: ε

text: diverging

text: to

text: as

text: ε

text: such

text: that,

text: for

text: all

text: t

text: T

text: ε

text: the

text: solution

text: v

text: to

text: satisﬁes

text: v

text: ∞(t)

text: C

text: Ωε

text: ∞t

text: e

text: µεt

text: L

text: L

text: L

text: for

text: some

text: positive

text: constant

text: C

text: and

text: µε

text: sup

text: λε(ξ)

text: C

text: Ωε

text: L

text: ξ

text: The

text: proof

text: of

text: Theorem

text: we

text: present

text: here

text: is

text: based

text: on

text: the

text: theory

text: of

text: stable

text: families

text: of

text: generators,

text: ﬁrstly

text: developed

text: by

text: Pazy

text: in

text: it

text: is

text: a

text: generalization

text: of

text: the

text: theory

text: of

text: semigroups

text: for

text: evolution

text: systems

text: of

text: the

text: form

text: u

text: Lu,

text: when

text: the

text: linear

text: operator

text: L

text: t

text: depends

text: on

text: time.

text: We

text: refer

text: to

text: Appendix

text: A

text: for

text: the

text: deﬁnitions

text: of

text: the

text: tools

text: we

text: shall

text: use

text: in

text: the

text: following.

text: M

text: Proof

text: of

text: Theorem

text: First

text: of

text: all,

text: let

text: us

text: notice

text: that

text: ε

text: is

text: a

text: bounded

text: operator

text: that

text: sat-

text: ξ

text: isﬁes

text: the

text: estimate

text: εξ

text: θε(ξ)

text: Ωε

text: L∞,

text: ξ

text: I.

text: and

text: Hε(ξ)

text: is

text: such

text: that

text: Hε

text: Ωε

text: L

text: L

text: for

text: some

text: positive

text: constants

text: c

text: and

text: c

text: Moreover,

text: concerning

text: the

text: nonlinear

text: terms

text: ρε

text: and

text: R

text: ε

text: and

text: because

text: of

text: the

text: speciﬁc

text: form

text: of

text: ε,

text: there

text: follows

text: |R

text: ε

text: C

text: v

text: L

text: L

text: L

text: M

text: Next,

text: we

text: want

text: to

text: show

text: that

text: ε

text: ε

text: is

text: the

text: inﬁnitesimal

text: generator

text: of

text: a

text: C

text: semigroup

text: T

text: ξ

text: ξ

text: L

text: (t,

text: s).

text: To

text: this

text: aim,

text: concerning

text: the

text: eigenvalues

text: of

text: the

text: linear

text: operator

text: ε,

text: we

text: know

text: that

text: ξ

text: ξ

text: λε(ξ)

text: is

text: negative

text: and

text: goes

text: to

text: zero

text: as

text: ε

text: for

text: all

text: ξ

text: I.

text: Hence,

text: deﬁning

text: Λε

text: sup

text: λε(ξ),

text: ξ