Nonexistence of global solutions to wave Equations with structural
damping and nonlinear memory
Abstract
For the following wave equations with structural damping and nonlinear
memory source terms \[
u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u
+(-\Delta)^{\frac{\beta}{2}}u_t
=\int_{0}^{t}(t-s)^{\gamma-1}
\vert u
(s)\vert^{p}\,\text{d}s,
\] and \[
u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u
+(-\Delta)^{\frac{\beta}{2}}u_t
=
\int_{0}^{t}(t-s)^{\gamma-1}
\vert u_s
(s)\vert^{p}\,\text{d}s,
\] posed in $(x,t) \in
\mathbb{R}^N \times
[0,\infty) $, where $u=u(x,t)$ is real-value unknown
function, $p>1$,
$\alpha,\beta\in (0,
2]$, $\gamma\in (0,1)$, we prove the
nonexistence of global solutions. Moreover, we give an upper bound
estimate of the life span of solutions.
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