Traveling fronts of a conservation law with hyper-dissipation

Abstract

Studied here are traveling-front solutions $\phi_{\epsilon}(x - {c} t)$ of a conservation law with hyper-dissipation appended. The evolution equation in question is a simple conservation law with a fourth-order dissipative term, namely $$u_t + 2uu_x + {\epsilon} u_{xxxx} = 0 , $$ where ${\epsilon} > 0$. The traveling front is restricted by the asymptotic conditions $\phi_{\epsilon}(x) \to L_\pm$ as $x \to \pm\infty$, where $L_+ < L_-$, and the symmetry condition $\phi_{\epsilon}(x) + \phi_{\epsilon}(-x) = L_- + L_+$ for all $x \in \mathbb R$. Such fronts are shown to exist and proven to be unique. Unlike the corresponding fronts for the Burgers' equation, they do not decay monotonically to their asymptotic states, but oscillate infinitely often around them. Despite this oscillation, it is also shown that $\phi_{\epsilon}(x) \to L_+$ as ${\epsilon} \to 0$, for all $x > 0$, and $\phi_{\epsilon}(x) \to L_-$ as ${\epsilon} \to 0$, for all $x < 0$.