Convergence rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg–de Vries–Burgers equations

Abstract

In this paper we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg–de Vries–Burgers equations (E) u t + δ u x x x − ν u x x + f ( u ) x = 0 , t > 0 , x ∈ R with prescribed initial data (I) u ( x , 0 ) = u 0 ( x ) → u ± , as x → ± ∞ . Here δ ≠ 0 and ν > 0 are real constants, u + ≠ u − are two given constants and the smooth nonlinear function f ( u ) is assumed to be either convex or concave. An exponential time decay rate of its global solution toward traveling wave solutions is obtained by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura [S. Kawashima, A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101, 1985 97–127] and later elaborated by Matsumura, Nishihara [A. Matsumura, K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994), 83–96] and Nishikawa [M. Nishikawa, Convergence rates to the traveling wave for viscous conservation laws. Funkcial. Ekvac. 41 (1998), 107–132].