Convergence to shock profiles for Burgers equation with singular fast-diffusion and boundary effect

Abstract

In this paper, we study the asymptotic stability of viscous shock profile for the Burgers equation $u_t+f(u{)}_x=(\frac{u_x}{u^{1-m}}{)}_x$ on the half-space $(0,+\infty )$, subject to the boundary conditions $u{|}_{x=0}=u_{-}>0$ and $u{|}_{x=+\infty }=0$. Here, the parameter $\frac{1}{2}<m<1$ measures the strength of fast diffusion. A key challenge arises from the pronounced singularity in the diffusivity ${\left(\frac{u_x}{u^{1-m}}\right)}_x$ at $u=0$ and the boundary layer. We demonstrate that the long-time behavior of $u$ converges to a shifted shock profile $U(x-st-d(t))$, where $d(t)$ is governed by the boundary layer dynamics at $x=0$ and driven by the initial data $u(x,0)$. To overcome the singularity from fast diffusion compounded by the bad effect of boundary layer for wave stability, some new techniques for weighted energy estimates are introduced artfully.