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Blow-up for semilinear wave equations with time-dependent damping in an exterior domain

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    * Corresponding author
B. Samet is supported by Researchers Supporting Project RSP-2019/4, King Saud University, Riyadh, Saudi Arabia
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  • We consider the semilinear wave equation with time-dependent damping

    $ \partial_{tt}u-\Delta u +\mu (1+t)^{-\beta} \partial_t u = |u|^p, \quad (t, x)\in (0, \infty)\times D^c, $

    where $ D^c = \mathbb{R}^N\backslash D $, $ D $ is the closed unit ball in $ \mathbb{R}^N $, $ N\geq 2 $, $ \mu>0 $, $ p>1 $ and $ -1<\beta<1 $. The considered equation is investigated under the boundary conditions:

    $ u(t, x) \left(\mbox{or } \frac{\partial u}{\partial n^+}(t, x)\right) = b(t)f(x)\, \, \mbox{on}\, \, (0, \infty)\times \partial D, $

    where $ n^+ $ is the outward (relative to $ D^c $) unit normal on $ \partial D $. General blow-up results are established for the considered problems. Moreover, for a certain class of functions $ b $, the critical exponent in the sense of Fujita is obtained.

    Mathematics Subject Classification: Primary: 35B33, 35L71; Secondary: 35L05.


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