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Upper and Lower Bounds of Blowup Time to a Strongly Damped Wave Equation of Kirchhoff Type with Memory Term and Nonlinear Dissipations

  • Amir Peyravi
  • Faramarz Tahamtani
Article
  • 73 Downloads

Abstract

In this paper, a wave equation of Kirchhoff type of the form
$$\begin{aligned}&u_{tt}-M(\Vert \nabla u\Vert _{2}^{2})\Delta u + \int _{0}^{t}g(t-s)\Delta u(s)\mathrm{d}s+h(u_{t})-\mu \Delta u_{t}\\&\quad =|u|^{\alpha }u,\quad \mathrm {in}\quad \Omega \times (0,+\infty ) \end{aligned}$$
is considered. Under suitable assumptions on the viscoelastic term and initial data, we prove that the solutions blow up at a finite time. Our result improves the previous work by Chen and Liu (IMA J Appl Math 1–29, 2015) in which the authors obtained an exponential growth of solutions where the memory kernel depends on initial energy and we show that it is not necessary to have such a restriction. This also extends the work by Hu et al. (Bound Value Probl 2017:112, 2017) where the relaxation function has not been considered. Some estimates for lower bounds of blowup time are also given.

Keywords

Kirchhoff equation blowup strong damping nonlinear dissipation viscoelastic 

Mathematics Subject Classification

35L20 35B44 45K05 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShiraz UniversityShirazIran

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