Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

Abstract

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations $$u_t- u_{txx}- vu_{xx}+\beta u_x+f(u)_x = 0,\, t > 0,\, x \in {\bf R}\quad\quad ({\rm E})$$ with prescribed initial data $$u(x, 0) = u_0(x)\rightarrow u_{\pm},\quad {\rm as}\,x\rightarrow\pm\infty.\quad\quad ({\rm I})$$ Here v( > 0), β are constants, u ± are two given constants satisfying u+ ≠ u− and the nonlinear function f(u) ∈C2(R) is assumed to be either convex or concave. An algebraic time decay rate to traveling waves of the solutions of the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equation is obtained by employing the weighted energy method developed by Kawashima and Matsumura in [6] to discuss the asymptotic behavior of traveling wave solutions to the Burgers equation.