APPROXIMATION OF SOLUTIONS OF THE ROBIN-DIRICHLET PROBLEMS FOR DAMPED WAVE EQUATIONS

Abstract

In this paper, we investigate the Robin-Dirichlet problems (P_n) for dampedwave equations with arithmetic-mean terms

(S_{n}u)(t)=\dfrac{1}{n}\sum\limits_{i=1}^{n}u^{2}(\tfrac{i-1}{n},t), and

(Ŝ_{n}u)(t)=\dfrac{1}{n}\sum\limits_{i=1}^{n}u_{x}^{2}(\tfrac{i-1}{n},t),

where u is the unknown function. First, under suitable conditions, weprove that, for each n ∈ ℕ, (P_n) has a unique weak solution ū_n Next, we prove that thesequence of solutions un converge strongly in appropriate spaces to the weak solution u_∞ of the corresponding problem (P_∞). Some remarks on open problems are also given in theend of paper.