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General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memory terms. (English) Zbl 1441.35041

Summary: In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation
\[\begin{aligned}u_{tt}-\Delta u+u+(g*\Delta u)(t)+h(u_t)u_t+|u|^2u=u\log |u|^k,\end{aligned}\]
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 Q. Hu et al. [ibid. 79, No. 1, 131–144 (2019; Zbl 1415.35043)] 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.

MSC:

35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L71 Second-order semilinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
74D10 Nonlinear constitutive equations for materials with memory
93D20 Asymptotic stability in control theory

Citations:

Zbl 1415.35043
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References:

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