TY - JOUR

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

AU - Jendoubi, M. A.

AU - Poláčik, P.

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.

UR - http://www.scopus.com/inward/record.url?scp=0344118917&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0344118917&partnerID=8YFLogxK

U2 - 10.1017/s0308210500002845

DO - 10.1017/s0308210500002845

M3 - Article

AN - SCOPUS:0344118917

SN - 0308-2105

VL - 133

SP - 1137

EP - 1153

JO - Royal Society of Edinburgh - Proceedings A

JF - Royal Society of Edinburgh - Proceedings A

IS - 5

ER -