Large-time behavior of solutions to certain quasilinear parabolic equations in several space dimensions

Abstract:

We consider the Cauchy problem, ${u_t} + {\text {div}}f(u) = \Delta u$ for $x \in {{\mathbf {R}}^n},t > 0$ with $u(x,0) = {u_0}(x)$. For $n = 1$, suppose $f'' > 0$ and $\smallint \left | {{u_0} - \phi } \right |dx < \infty$ where $\phi$ is piecewise constant and $\phi (x) \to {u^ + }({u^ - })$ as $x \to + \infty ( - \infty )$. A result of Il’in and Oleinik states that if $\phi (x - kt)$ is an entropy solution of ${u_t} + {\text {div}}f(u) = 0$, then $u(x,t)$ approaches a traveling wave solution, $\tilde u(x - kt)$, as $t \to \infty$, with $\tilde u(x) \to {u^ + }({u^ - })$ as $x \to + \infty ( - \infty )$. We give two examples which show that this result does not hold for $n \geqslant 2$.