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

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15 pages
English

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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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Publié le 01 janvier 2012
Nombre de lectures 33
Langue English

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Tahamtani and ShahrouziBoundary Value Problems2012,2012:50 http://www.boundaryvalueproblems.com/content/2012/1/50
R E S E A R C HOpen Access Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term * Faramarz Tahamtaniand Mohammad Shahrouzi
* Correspondence: tahamtani@shirazu.ac.ir Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71454, Iran
Abstract We consider the semilinear Petrovsky equation t 2 2p tt+Δug(ts)Δu(s)ds=|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. Keywords:viscoelasticity, existence, blowup, lifespan, negative initial energy, posi tive initial energy
1 Introduction In this article, we concerned with the problem 2 2p utt+Δug(ts)Δu(s)ds=|u|u,xΩ,τ > 0 (1:1) u(x,t) =νu(x,t) = 0,x∂Ω,t0 u x, 0=u0x,utx=, 0u1x,xΩ n whereΩRis a bounded domain with smooth boundaryΩin order that the divergence theorem can be applied.νis the unit normal vector pointing toward the exterior ofΩandp> 0. Here,grepresents the kernel of the memory term satisfying some conditions to be specified later. In the absence of the viscoelastic term, i.e., (g= 0), we motivate our article by pre senting some results related to initialboundary value Petrovsky problem 2 utt+Δu=f(u,ut),xΩ,t>0 u(x,t) =νu(x,t) = 0,x∂Ω,t0(1:2) u x, 0=u0x,utx=, 0u1x,xΩ
© 2012 Tahamtani and Shahrouzi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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