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General Stability and Exponential Growth for a Class of Semi-linear Wave Equations with Logarithmic Source and Memory Terms

  • Amir Peyravi
Article
  • 33 Downloads

Abstract

In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation
$$\begin{aligned} u_{tt}-\Delta u + u + (g\,*\, \Delta u)(t)+ h(u_{t})u_{t}+|u|^{2}u=u\log |u|^{k}, \end{aligned}$$
in an open bounded domain \(\Omega \subseteq \mathbb {R}^3\) whith \(h(s)=k_{0}+k_{1}|s|^{m-1}.\) We prove a general stability of solutions which improves and extends some previous studies such as the one by Hu et al. (Appl Math Optim,  https://doi.org/10.1007/s00245-017-9423-3) in the case \(g=0\) and in presence of linear frictional damping \(u_{t}\) when the cubic term \(|u|^2u\) is replaced with u. In the case \(k_{1}=0,\) we also prove that the solutions will grow up as an exponential function. Our result shows that the memory kernel g dose not need to satisfy some restrictive conditions to cause the unboundedness of solutions.

Keywords

General decay Exponential growth Logarithmic wave equations Memory damping 

Mathematics Subject Classification

35B35 35B40 74D10 93D20 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

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

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