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On the decay of the energy for systems with memory and indefinite dissipation. (English) Zbl 1115.35024

Summary: We study the asymptotic behavior of the viscoelastic system with nondissipative kernels. We show that the uniform decay of the energy depends on the decay of the kernel, the positivity of the kernel in \(t=0\) and some smallness condition. That is, if the kernel \(g\in C^2(\mathbb{R}^+)\) with \(g(0)>0\), decays exponentially to zero then the solution decays exponentially to zero. On the other hand, if the kernel decays polynomially as \(t^{-p}\) then the corresponding solutions also decays polynomially to zero with the same rate decay.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L90 Abstract hyperbolic equations
74D05 Linear constitutive equations for materials with memory
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