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Almost-periodic forcing for a wave equation with a nonlinear, local damping term

Published online by Cambridge University Press:  14 November 2011

Alain Haraux
Alain Haraux
Affiliation:
Analyse Numerique, Tour 55–65, 5e étage, Université Pierre et Marie Curie, 4, place Jussieu, 75230 Paris Cedex 05, France
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Synopsis

Let Ω⊂ℝnbe a bounded open domain and T = ∂Ω. It β is a maximal monotone graph in ℝ×ℝ with 0ϵβ(0), and f: ℝ×Ω→ℝ is measurable with t→ f(t,.) S2-almost periodic as a function ℝ→L2(Ω), we consider the nonlinear hyperbolic equation

We show that:

  • (i) if ゲ is strictly increasing and (1) has a solution ω on ℝ with [ω, Əω/Ət] almost periodic: , for any solution of (1) there exists with u(t,.)–ω(t,.)—ξin

  • (ii) if β is single valued and everywhere defined, the existence of ω as above implies that, for every solution of (1), there exists Ϛ(t, x) with ә2Ϛ/әt2–0△Ϛ = in ℝ×Ω and u(t,.)–ω(t,.)—0 in as t → +∞

  • (iii) if β–1 is uniformly continuous and ゲ satisfies some growth assumption (depending on N), for every f as above, there exists ω solution of (1) on ℝ with [ω, Əω/Ət] almost periodic: ℝ → .

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 3-4 , 1983 , pp. 195 - 212
Copyright
Copyright © Royal Society of Edinburgh 1983

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