- Results in Nonlinear Analysis
- Vol: 5 Issue: 2
- Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmeti...
Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms
Authors : Le Thi Phuong Ngoc, Nguyen Vu Dzung, Nguyen Huu Nhan, Nguyen Thanh Long
Pages : 191-212
Doi:10.53006/rna.1082465
View : 13 | Download : 8
Publication Date : 2022-06-30
Article Type : Research
Abstract :In this paper, we study the Robin-Dirichlet problem $(P_{n})$ for a strongly damped wave equation with arithmetic-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.Keywords : Robin-Dirichlet problem, Arithmetic-mean terms, Faedo-Galerkin method, Linear recurrent sequence