Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation
- 1 June 1980
- journal article
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 32 (3) , 631-643
- https://doi.org/10.4153/cjm-1980-049-5
Abstract
In this paper we study the nonlinear initial boundary value problem(1.1) ωtt— αΔ ωt— Δω= f(ω), t> 0ω(x, 0) = ϕ(x), x∈ Ωωt(x, 0) = ψ (x), x∈ Ωω(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.Keywords
This publication has 1 reference indexed in Scilit:
- Nonlinear semigroups and differential equations in Banach spacesPublished by Springer Nature ,1976