Global Existence And Optimal Time Decay Estimate For A Type Of Wave Equation With Damping

In this paper we focus on the global existence and optimal time decay estimate ofsolution of nonlinear wave equation with damping. Concretely speaking, we considerthe half space problem with zero boundary condition and the Cauchy problem withnonlinear memory. We first study the time decay estimate of solution operator of linearpart in half space. Next we deifne a reasonable function norm and weighted Sobolevspace to consider the estimate of nonlinear part. Finally, We construct a transform byDuhamel principle and prove the global existence and time decay estimate of solutionunder the condition of small initial data. With the same theoretical framework weconsider the global existence in the whole space with nonlinear memory.