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

Abstract

In this paper, we study the Robin-Dirichletproblem $(P_{n})$ for a strongly damped wave equation witharithmetic-mean terms $S_{n}u$ and $\hat{S}_{n}u,$ where $u$ is the unknown function, $S_{n}u=\tfrac{1}{n}\sum\nolimits_{i=1}^{n}u(\tfrac{i-1}{n},t)$ and $\hat{S}_{n}u=\tfrac{1}{n}\sum\nolimits_{i=1}^{n}u_{x}^{2}(\tfrac{i-1}{n},t)$. First, under suitable conditions, we prove that, for each $n\in \mathbb{N},$ $(P_{n})$ has a unique weak solution $u^{n}$. Next, we prove that the sequence of solutions $u^{n}$ converge strongly in appropriate spaces to the weak solution $u$ of the problem $(P),$ where $(P)$ is defined by $(P_{n})$ in which the arithmetic-mean terms $S_{n}u$ and $\hat{S}_{n}u$ are replaced by $\int\nolimits_{0}^{1}u(y,t)dy$ and$\int\nolimits_{0}^{1}u_{x}^{2}(y,t)dy,$ respectively. Finally,some remarks on a couple of open problems are given.