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

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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.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71467-13565, Iran

    Amir Peyravi

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  1. Amir Peyravi
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Peyravi, A. General Stability and Exponential Growth for a Class of Semi-linear Wave Equations with Logarithmic Source and Memory Terms. Appl Math Optim 81, 545–561 (2020). https://doi.org/10.1007/s00245-018-9508-7

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