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.

中文翻译：

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一类具有对数源和记忆项的半线性波动方程的一般稳定性和指数增长

在这项工作中，我们研究对数波动方程$$ \ begin {aligned} u_ {tt}-\ Delta u + u +（g \，* \，\ Delta u）（t）的无穷大处的渐近稳定性和不稳定性+ h（u_ {t}）u_ {t} + | u | ^ {2} u = u \ log | u | ^ {k}，\ end {aligned} $$在开放式边界域\（\ Omega \ subseteq \ mathbb {R} ^ 3 \）均可进行\（H（S）= K_ {0} + K_ {1} | S | ^ {M-1} \）。我们证明解决方案的一般稳定性从而提高和延伸之前的一些研究，例如Hu等人的研究。（Appl Math Optim，https：//doi.org/10.1007/s00245-017-9423-3）在\（g = 0 \）的情况下，并且在存在线性摩擦阻尼\（u_ {t} \）的情况下三次项\（| u | ^ 2u \）替换为u。在这种情况下\（k_ {1} = 0，\）我们还证明了解将作为指数函数增长。我们的结果表明，存储核g无需满足某些限制性条件即可导致解的无穷大。