Global stability of stationary waves for damped wave equations

Abstract

Nonlinear stability of stationary waves to damped wave equations hasbeen studied by many authors in recent years, the main difficultylies in how to control the possible growth of its solutions causedby the nonlinearity of the equation under consideration. Aneffective way to overcome such a difficulty is to use the smallnessof the initial perturbation and/or the smallness of the strength ofthe stationary waves and based on this argument, local stability ofstrong increasing stationary waves for convex flux functions andweak decreasing stationary waves for general flux functions arewell-established, cf. [11,13,15]. As to the nonlinear stability resultwith large initial perturbation, the only results available now are[3,4] which use the smallness ofthe stationary waves to overcome the above mentioned difficulty andconsequently, although the initial perturbation can be large, itdoes require that it satisfies certain growth condition as thestrength of the stationary waves tends to zero. Thus a naturalquestion of interest is, for any initial perturbation lying incertain Sobolev space $H^s\left({\bf R}_+\times{\bfR}^{n-1}\right)$, how to obtain the global stability of stationarywaves to the damped wave equations? The main purpose of thismanuscript is devoted to this problem.