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Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation

Published online by Cambridge University Press:  20 November 2018

G. F. Webb
G. F. Webb*
Affiliation:
Vanderbilt University, Nashville, Tennessee
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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 Rn, n = 1, 2, 3, α > 0, and f ∈ C1(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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 631 - 643
Copyright
Copyright © Canadian Mathematical Society 1980

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