Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping

M. A. Jendoubi, P. Poláčik

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Abstract

We consider two types of equations on a cylindrical domain Ω × (0, ∞), where Ω is a bounded domain in ℝN, N ≥ 2. The first type is a semilinear damped wave equation, in which the unbounded direction of Ω × (0, ∞) is reserved for time t. The second type is an elliptic equation with a singled-out unbounded variable t. In both cases, we consider solutions that are defined and bounded on Ω × (0, ∞) and satisfy a Dirichlet boundary condition on ∂Ω × (0, ∞). We show that, for some nonlinearities, the equations have bounded solutions that do not stabilize to any single function φ: Ω → ℝ, as t → ∞; rather, they approach a continuum of such functions. This happens despite the presence of damping in the equation that forces the t derivative of bounded solutions to converge to 0 as t → ∞. Our results contrast with known stabilization properties of solutions of such equations in the case N = 1.

Original languageEnglish (US)
Pages (from-to)1137-1153
Number of pages17
JournalRoyal Society of Edinburgh - Proceedings A
Volume133
Issue number5
DOIs
StatePublished - 2003

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