TY - JOUR

T1 - Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping

AU - Jendoubi, M. A.

AU - Poláčik, P.

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

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U2 - 10.1017/s0308210500002845

DO - 10.1017/s0308210500002845

M3 - Article

AN - SCOPUS:0344118917

VL - 133

SP - 1137

EP - 1153

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 5

ER -