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We consider an initial boundary value problem for the equation $u_{t t} - Δ u - \nabla ϕ \cdot \nabla u + f (u) + g (u_{t}) = 0$ . We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

Decay of positive waves in nonlinear systems of conservation laws

Alberto Bressan, Rinaldo M. Colombo (1998)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Decay of solutions of a degenerate hyperbolic equation.

Dix, Julio G. (1998)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Decay of solutions of a nonlinear hyperbolic system in noncylindrical domain.

Nunes Rabello, Tania (1994)

International Journal of Mathematics and Mathematical Sciences

Decay of solutions of a system of nonlinear Klein-Gordon equations.

Ferreira, José, Perla Menzala, Gustavo (1986)

International Journal of Mathematics and Mathematical Sciences

Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type

Barbara Szomolay (2003)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the asymptotic behavior of solutions to the damped, nonlinear vibration equation with self-interaction $\ddot{u} = - γ \dot{u} + {m (∥ \nabla u ∥}^{2} {) Δ u - δ | u |}^{α} u + f,$ which is known as degenerate if $m (\cdot) \geq 0$ , and non-degenerate if $m (\cdot) \geq m_{0} > 0$ . We would like to point out that, to the author’s knowledge, exponential decay for this type of equations has been studied just for the special cases of $α$ . Our aim is to extend the validity of previous results in [5] to $α \geq 0$ both to the degenerate and non-degenerate cases of $m$ . We extend our results to equations with...

Decay of solutions of the elastic wave equation with a localized dissipation

Mourad Bellassoued (2003)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Decay of solutions of the wave equation in the exterior of several convex bodies

Mitsuru Ikawa (1988)

Annales de l'institut Fourier

We study the decay of solutions to the wave equation in the exterior of several strictly convex bodies. A sufficient condition for exponential decay of the local energy is expressed in terms of the period and the Poincare map of periodic rays in the exterior domain.

Decay rates for solutions of a system of wave equations with memory.

de Lima Santos, Mauro (2002)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Decay rates for solutions of a Timoshenko system with a memory condition at the boundary.

De Lima Santos, Mauro (2002)

Abstract and Applied Analysis

Decay rates for solutions of damped systems and generalized Fourier transforms.

Reinhard Racke (1990)

Journal für die reine und angewandte Mathematik

Decay rates for solutions of semilinear wave equations with a memory condition at the boundary.

de Lima Santos, Mauro (2002)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Decay rates of Volterra equations on ℝN

Monica Conti, Stefania Gatti, Vittorino Pata (2007)

Open Mathematics

This note is concerned with the linear Volterra equation of hyperbolic type $\partial_{t t} u (t) - α Δ u (t) + \int_{0}^{t} μ (s) Δ u (t - s) d s = 0$ on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.

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