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

Tahamtani Faramarz, Shahrouzi, Shahrouzi Mohammad

Description

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.

Sujets

• Viscoelasticity
• Existence
• Blow-Up
• Life expectancy

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 33
• Langue: English

Extrait

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: [e-mail] Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71454, Iran

Abstract We consider the semilinear Petrovsky equation t  2 2p tt +Δu−g(t−s)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 u tt +Δu−g(t−s)Δu(s)ds= | u | u,x∈Ω,τ > 0 (1:1) u(x,t) =∂ ν u(x,t) = 0,x∈∂Ω,t≥0 u x, 0=u 0 x,u t x=, 0u 1 x,x∈Ω n whereΩ⊂Ris a bounded domain with smooth boundary∂Ωin 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 u tt +Δ u =f(u,u t ),x∈Ω,t>0 u(x,t) =∂ ν u(x,t) = 0,x∈∂Ω,t≥0 (1:2) u x, 0=u 0 x,u t x=, 0u 1 x,x∈Ω

© 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.