TITLE: Boundary and initial conditions in quasi linear first order pde
QUESTION [7 upvotes]: I cannot understand what we are looking to find in such a problem... For example consider the pde 
$u_t+u_x=u$, with $x,t>0$ (1)
and initial and boundary conditions: 
$u(x,0)=1$, for $x\ge0$  (2)
$u(0,t)=1$, for $t\ge0$  (3)
Are we looking for a solution of (1) which can be extended (continously?)in order to take values implemented by (2), (3)?
Remark: I do not need a solution of the above problem, just a proper defintion or refference. Thanks in advance.

REPLY [2 votes]: Are we looking for a solution of (1) which can be extended
  (continously?)in order to take values implemented by (2), (3)?

Yes (and yes, continously).