Global Existence And Convergence Rates Of Solutions Of Quasilinear Hyperbolic Equations With Time-dependent Damping

We study the existence and convergence rates of global smooth solutions to the Cauchy problem of the quasi-linear hyperbolic equations with time-dependent damping.Regarding the study of damped quasi-linear hyperbolic equations,previous results mainly focus on the case of constant damping.This article mainly studies timedependent damping,the damping term is ?/((1+t)?)Vt,where-1<?<1,?>0 is a constant.When ?>0,the damping effect of the damping term ?/((1+t)?)Vt is timegradually-degenerate,when-1<?<0,the damping effect of the damping term ?/((1+t)?)Vt is time-gradually-enhancing.We use the energy methods and the timeweighted energy method to prove the global existence of smooth solutions of the quasi-linear hyperbolic equations with time-dependent damping,and get the convergence rates of the solutions.The paper is organized as follows:In Chapter One,introduces the current research status of quasi-linear equations with time-dependent damping,and summarizes the main results,content arrangement and preliminary knowledge of this article.In Chapter Two,the existence and decay estimate of the global solution of the quasi-linear equation with time-dependent damping are proved.