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2013 ASYMPTOTIC BEHAVIOR AND BLOW-UP OF SOLUTIONS FOR A NONLINEAR VISCOELASTIC WAVE EQUATION WITH BOUNDARY DISSIPATION
Faramarz Tahamtani, Amir Peyravi
Taiwanese J. Math. 17(6): 1921-1943 (2013). DOI: 10.11650/tjm.17.2013.3034

Abstract

We study the nonlinear viscoelastic wave equation $$ u_{tt} - k_{0} \Delta u + \int_{0}^{t} g(t-s) \mathrm{div} \bigr[a(x) \nabla u(s) \bigr] ds + \bigr(k_{1} + b(x) |u_{t}|^{m-2}\bigr) u_{t} = |u|^{p-2}u $$ with dissipative boundary conditions. Under some restrictions on the initial data and the relaxation function and without imposing any restrictive assumption on $a(x)$, we show that the rate of decay is similar to that of $g$. We also prove the blow-up results for certain solutions in two cases. In the case $k_{1} = 0$, $m=2$, we show that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy. In another case, $k_{1}\geq 0$, $m\geq 2$, we prove a nonexistence result when the initial energy is less than potential well depth.

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Faramarz Tahamtani. Amir Peyravi. "ASYMPTOTIC BEHAVIOR AND BLOW-UP OF SOLUTIONS FOR A NONLINEAR VISCOELASTIC WAVE EQUATION WITH BOUNDARY DISSIPATION." Taiwanese J. Math. 17 (6) 1921 - 1943, 2013. https://doi.org/10.11650/tjm.17.2013.3034

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1286.35152
MathSciNet: MR3141867
Digital Object Identifier: 10.11650/tjm.17.2013.3034

Subjects:
Primary: 35B40 , 35B44 , 35L20 , 35L70

Keywords: asymptotic behavior , Blow-up , boundary dissipation

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

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Vol.17 • No. 6 • 2013
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