Abstract
In this article we propose approximation schemes for solving nonlinear initial boundary value problem with Volterra operator. Existence, uniqueness of solution as well as some regularity results are obtained via Rothe-Galerkin method. Mathematics Subject Classification 2000: 35k55; 35A35; 65M20.
Highlights
1 Introduction The aim of this work is the solvability of the following equation
The memory operator K is defined by t
We prove that a(umn ) → a(u)in Lp((0, T), W01,p( ))
Summary
Existence of solutions for a class of nonlinear evolution equations of second order is proved by studying a full discretization. 2 Hypothesis and mean results To solve problem (P), we assume the following hypotheses: (H1) The function b : R ® R is continuous, nondecreasing, b (0) = 0, b (u0) Î L2 (Ω) and satisfies |b(s)|2 ≤ C1B* (a (s)) + C2, ∀s Î R. Theorem 2 Under hypotheses (H1) - (H6), there exists a weak solution u for problem (P) in the sense of Definition 1.
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