Abstract
The purpose of this paper is to show that the set of weak solutions of the initial-boundary value problem for the linear wave equation is nonempty, connected, and compact.
Highlights
In this paper, we consider the following problem: find a pair (u,P) of functions satisfying utt − uxx + Ku + λut = F(x, t), 0 < x < 1, 0 < t < T, ux(0, t) = P(t), u(1, t) = 0, u(x, 0) = u0(x), ut(x, 0) = u1(x), (1.1)where the constants K, λ, the functions u0, u1, F are given before satisfying conditions specified later, and the unknown function u(x,t) and the unknown boundary value P(t) satisfy the following integral equation: P(t) = g(t) + K1 u(0, t) α−2u(0, t) + λ1 ut(0, t) β−2ut(0, t) −t k(t − s)u(0,s)ds, (1.2)in which the constants K1, λ1, α, β and the functions g, k are given before
Uc 1 < M for all c ∈ S, this means that US ⊂ S. (b) we show that U : S → Y is continuous
(c) we show that U(S) is relatively compact in Y
Summary
We consider the following problem: find a pair (u,P) of functions satisfying utt − uxx + Ku + λut = F(x, t), 0 < x < 1, 0 < t < T, ux(0, t) = P(t), u(1, t) = 0, u(x, 0) = u0(x), ut(x, 0) = u1(x),. Where the constants K, λ, the functions u0, u1, F are given before satisfying conditions specified later, and the unknown function u(x,t) and the unknown boundary value P(t) satisfy the following integral equation: P(t) = g(t) + K1 u(0, t) α−2u(0, t) + λ1 ut(0, t) β−2ut(0, t) −. In which the constants K1, λ1, α, β and the functions g, k are given before. This paper is a continuation of authors’ series of papers dealing with mixed problems for wave equations; see for instance the papers [1–5] among many others. The problem (1.1)-(1.2) is the mathematical model describing a shock problem involving a linear viscoelastic bar [1, 2, 4]
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