Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

Abstract

In this paper, we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations (E) u t − u t x x − ν u x x + β u x + f ( u ) x = 0 , t > 0 , x ∈ R with prescribed initial data (I) u ( 0 , x ) = u 0 ( x ) → u ± , as x → ± ∞ . Here ν ( > 0 ) , β ∈ R are constants, u ± are two given constants satisfying u + ≠ u − and the nonlinear function f ( u ) ∈ C 2 ( R ) is assumed to be either convex or concave. Based on the existence of traveling waves, the local stability and the algebraic decay rate to traveling waves of solutions to the Cauchy problem (E) and (I) established in Yin et al. (2007) [13], we show an exponential decay rate of the solutions to the Cauchy problem (E) and (I) toward the traveling waves mentioned above, by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura in (1985) [14] and later elaborated by Matsumura and Nishihara (1994) [15] and Nishikawa (1998) [16].