The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation
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March 2014, 13(2): 835-858. doi: 10.3934/cpaa.2014.13.835
Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux
Tong Li 1, and Hui Yin 2,
Department of Mathematics, University of Iowa, Iowa City, IA 52242
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Received June 2013 Revised August 2013 Published October 2013
This paper is concerned with the initial-boundary value problem for the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $R_+$ \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 nonlinear function $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b<0$, we show the existence and the 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.
Keywords: convergence rate., global stability, Generalized Benjamin-Bona-Mahony-Burgers equation, non-convex flux, boundary layer solution.
Mathematics Subject Classification: Primary: 35Q53, 35B35; Secondary: 76B1.
Citation: Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835
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