Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation

Abstract

In this paper we study the nonlinear initial boundary value problem (1.1) ω tt — αΔ ω t — Δω = f(ω), t > 0 ω(x, 0) = ϕ(x), x ∈ Ω ω t (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) ≦ c o for all x ∈ R (where c 0 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.