Dynamical behaviors of the solution to a periodic initial–boundary value problem of the generalized Rosenau-RLW-Burgers equation

Abstract

The behaviors of a solution to a periodic initial–boundary value problem (IBVP) for the generalized Rosenau-RLW-Burgers equation are analyzed in this paper. Under the smoothness of its initial value, the global existence and uniqueness of the solution to the periodic IBVP for the equation are proved by means of the continuation extension and L2-energy estimates. The impact of the viscous term on the equation for the behaviors of the global solution is theoretically investigated. More precisely, two types of nonlinear wave behaviors are dealt with; one is the exponential convergence of the global solution to the average of its initial value when the viscous term is nonzero, and the other is the oscillation of the global solution around the initial average at any given time when the viscous term vanishes. Additionally, numerical simulations are provided to illustrate and validate our theoretical results. Further, the effects of some parameters on the periodic IBVP for the generalized Rosenau-RLW-Burgers equation are discussed when proceeding with an initial Gaussian condition.