Abstract
In this article, we consider the nonlinear viscoelastic equation u t t - Δ u + ∫ 0 t g ( t - τ ) Δ u ( τ ) d τ - ω Δ u t + μ u t = u p - 2 u with initial conditions and Dirichlet boundary conditions. We first prove a local existence theorem and show, for some appropriate assumption on g and the initial data, that this solution is global with energy which decays exponentially under the potential well. Secondly, not only finite time blow-up for solutions starting in the unstable set is proved, but also under some appropriate assumptions on g and the initial data, a blow-up result with positive initial energy is established. Finally, we also prove the boundedness of global solutions for strong (ω > 0) damping case.2000 MSC: 35L05; 35L15; 35L70.
Highlights
In this article we study the behavior of solutions for the following nonlinear viscoelastic equation
L being the first eigenvalue of the operator -Δ under homogeneous Dirichlet boundary conditions, and
Using the ideas of the “potential well” theory introduced by Payne and Sattinger [28], we show that for some appropriate assumption on g and the initial data, that this solution is global with energy which decays exponentially under the potential well
Summary
They established a local existence result and showed, for certain initial data and suitable conditions on g, that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g. For the problem (1.4) in Rn and with m = 2, Kafini and Messaoudi [26] showed, for suitable conditions on g and initial data, that solutions with negative energy blow up in finite time.
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