A wave equation with structural damping and nonlinear memory
The author obtains a critical exponent for a wave equation with structural damping and nonlinear memory \(u_{tt}-\Delta u+\mu (-\Delta)^{1/2} u_t=\int_0^t (t-s)^{-\gamma} |u(s)|^p ds\), posed for \(x\in \mathbb{R}^n\), where \(\mu>0\) and \(\gamma\in (0,1)\). Let \(\bar p(n,\gamma):=\max(1+\frac{3-\gamma}{n+\gamma-2}, \frac{1}{\gamma})\) with \(n\geq 2\). It is proved that when \(p>\bar p(n,\gamma)\), we have global existence for sufficiently small data. On the other hand, in the special case that \(\mu=2\), it is proved that there is no global solutions for suitable arbitrarily small data, when \(1<p<\bar p(n,\gamma)\) for any \(n\geq 2\) and \(1<p<\infty\) for \(n=1\). This shows that the critical power for the problem to admit global small solutions is \(\bar p(n,\gamma)\).
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