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On a wave equation with supercritical interior and boundary sources and damping terms. (English) Zbl 1244.35092

$\left\{\begin{array}{c}{u}_{tt}-{\Delta }u+{g}_{0}\left({u}_{t}\right)={|u|}^{p-1}u\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }×\left[0,\infty \right),\hfill \\ {\partial }_{\nu }u+u+g\left({u}_{t}\right)={|u|}^{k-1}u\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma }×\left[0,\infty \right),\hfill \\ u\left(0\right)={u}_{0}\in {H}^{1}\left({\Omega }\right),\phantom{\rule{4pt}{0ex}}{u}_{t}\left(0\right)={u}_{1}\in {L}^{2}\left({\Omega }\right),\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$
where ${\Omega }\subset {ℝ}^{3}$ is a bounded open set with sufficiently smooth boundary ${\Gamma }$, the maps ${g}_{0}\left(s\right)$ and $g\left(s\right)$ are monotone and represent the interior and boundary dissipation, the Nemytski operators ${|u|}^{p-1}u$, $p>1$, ${|u|}^{k-1}u$, $k>1$, model the interior and boundary sources respectively.
The authors give conditions for ${g}_{0}$, $g$ and $p$, $k$ under which the problem (1) has unique global weak solution $u\in 𝒞\left(\left[0,\infty \right),{H}^{1}\left({\Omega }\right)\right)$. The exponential and algebraic uniform decay rates of the finite energy are established. The authors prove a blow up result for weak solutions of (1) with nonnegative initial energy.