TITLE: Nonlinear heat equation $u_{t} = \Delta(u^{4})$
QUESTION [9 upvotes]: Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that all solutions to this PDE tend to zero (pointwise) as $t \rightarrow \infty$.
The equation looks like the porous medium equation so I thought about looking at how the Barenblatt solution is derived, but the issue is that this doesn't cover all solutions, so I'm thinking that there is a general argument to prove the above result.

REPLY [4 votes]: Let $v = u^4$ and rewrite the PDE in terms of $v$ to get:
$$
v^{-3/4} v_t = \Delta v.
$$
Let $\;v = A(t) B(x,y,z)\;$ and separate variables to get:
$$
A' = 4 A^{7/4} c \quad \mbox{and} \quad B^{3/4} \Delta B = c \quad \mbox{for a constant } c .
$$
The DE in $A$ has the solution $\;A(t)^{3/4} = -\frac{4/3}{4 c t + k}\;$ for a constant $k$. So when $t$ goes to infinity $A$ goes to $0$ and therefore also $v$ and $u$.