Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation

Webb, G. F.

Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 631-643

Résumé

In this paper we study the nonlinear initial boundary value problem(1.1) ωtt — αΔ ωt — Δω= f(ω), t> 0 ω(x, 0) = φ(x), x∈ Ω ωt(x, 0) = ψ (x), x∈ Ω ω(x, t ) = 0, x ∈ ∂Ω, t ≥ 0.In (1.1) Ω is a smooth bounded domain in R n , n = 1, 2, 3, α> 0, and f ∈ C 1(R;R) with f‘(x) ≦ co for all x ∈ R (where c0 is a nonnegative constant), lim sup|x|→+∞f(x)/x ≦0, and f(0) = 0. Our objective will be to establish the existence of unique strong global solutions to (1.1) and investigate their behavior as t→ +∞.Our approach takes advantage of the semilinear character of (1.1) and reformulates the problem as an abstract ordinary differential equation in a Banach space.

Détail

DOI : 10.4153/CJM-1980-049-5

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