Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term

We consider the semilinear Petrovsky equation u t t + Δ 2 u - ∫ 0 t g ( t - s ) Δ 2 u ( s ) d s = u p u in a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with non-positive in...

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Published inBoundary value problems Vol. 2012; no. 1; pp. 1 - 15
Main Authors Tahamtani, Faramarz, Shahrouzi, Mohammad
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 26.04.2012
Hindawi Limited
BioMed Central Ltd
Subjects
Analysis
Approximations and Expansions
Boundary value problems
Classification
Difference and Functional Equations
Estimates
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinearity
Ordinary Differential Equations
Partial Differential Equations
existence
positive initial energy
negative initial energy
life-span
viscoelasticity
blow-up
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Summary:We consider the semilinear Petrovsky equation u t t + Δ 2 u - ∫ 0 t g ( t - s ) Δ 2 u ( s ) d s = u p u in a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with non-positive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given. Mathematics Subject Classification (2000) : 35L35; 35L75; 37B25.
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ISSN:1687-2770
1687-2762
1687-2770
DOI:10.1186/1687-2770-2012-50