Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux

Abstract

This paper is concerned with the initial-boundary value problem forthe generalized Benjamin-Bona-Mahony-Burgers equation in the halfspace $R_+$\begin{eqnarray}u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\u(t,0)=u_b.\end{eqnarray}Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$,$u_+\not=u_b$ are two given constant states and the nonlinearfunction $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b<u_+$. For the non-degenerate case $f'(u_+)<0$, we show the existence andthe global stability of strong boundary layer solution $\phi(x)$ with the above non-convex function $f(u)$ satisfying (1.3). In this case, the corresponding algebraic convergence rate is also obtained. Our analysis is based on the space-time weighted energy method (cf. [4, 11]) combined with some delicate estimates.