Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms

Abstract

In this paper, we study the Robin-Dirichlet problem (Pn) for a strongly damped wave equation with arithmetic-mean terms Snu and Sˆnu, where u is the unknown function, Snu =1nPni=1 u(i−1n, t) and Sˆnu =1 n Pni=1 u2x(i−1n, t). First, under suitable conditions, we prove that, for each n ∈ N, (Pn) has a unique weak solution un. Next, we prove that the sequence of solutions un converge strongly in appropriate spaces to the weak solution u of the problem (P), where (P) is defined by (Pn) in which the arithmetic-mean terms Snuand Sˆ nu are replaced by R 1 0 u(y, t)dy and R 1 0 u 2 x (y, t)dy, respectively. Finally, some remarks on a couple of open problems are given.