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 : 11 | Download : 10

Publication Date : 2022-06-30

Article Type : Research Paper

Abstract :In this paper, we study the Robin-Dirichlet problem $insert ignore into journalissuearticles values(P_{n});$ for a strongly damped wave equation with arithmetic-mean terms $S_{n}u$ and $\\hat{S}_{n}u,$ where $u$ is the unknown function, $S_{n}u=\\tfrac{1}{n} \\sum\\nolimits_{i=1}^{n}uinsert ignore into journalissuearticles values(\\tfrac{i-1}{n},t);$ and $\\hat{S}_{n}u= \\tfrac{1}{n}\\sum\\nolimits_{i=1}^{n}u_{x}^{2}insert ignore into journalissuearticles values(\\tfrac{i-1}{n},t);$. First, under suitable conditions, we prove that, for each $n\\in \\mathbb{N},$ $insert ignore into journalissuearticles values(P_{n});$ 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 $insert ignore into journalissuearticles values(P);,$ where $insert ignore into journalissuearticles values(P);$ is defined by $insert ignore into journalissuearticles values(P_{n});$ in which the arithmetic-mean terms $S_{n}u$ and $\\hat{S} _{n}u$ are replaced by $\\int\\nolimits_{0}^{1}uinsert ignore into journalissuearticles values(y,t);dy$ and $\\int\\nolimits_{0}^{1}u_{x}^{2}insert ignore into journalissuearticles values(y,t);dy,$ 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