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Existence and uniform decay for a non‐linear viscoelastic equation with strong damping
Author(s) -
Cavalcanti M. M.,
Domingos Cavalcanti V. N.,
Ferreira J.
Publication year - 2001
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.250
Subject(s) - viscoelasticity , mathematics , relaxation (psychology) , exponential growth , domain (mathematical analysis) , mathematical analysis , exponential decay , function (biology) , energy (signal processing) , energy method , physics , thermodynamics , statistics , quantum mechanics , psychology , social psychology , evolutionary biology , biology
This paper is concerned with the non‐linear viscoelastic equation$$|u_{t}|^{\rho} u_{tt} - \Delta u - \Delta u_{tt} + \int\nolimits_{0}^{t} g (t - \tau)\,\Delta u (\tau)\,{\rm{d}}\tau - \gamma \Delta u_{t} \,{=}\, 0$$We prove global existence of weak solutions. Furthermore, uniform decay rates of the energy are obtained assuming a strong damping Δ u t acting in the domain and provided the relaxation function decays exponentially. Copyright © 2001 John Wiley & Sons, Ltd.