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

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.