| Contents |
| The talk will deals with blow--up for the solutions of an evolution problem consisting on a semilinear wave equation posed in a bounded $C^{1,1}$ open subset of $I\!\!R^n$, supplied with a Neumann boundary condition involving a nonlinear dissipation.
The typical problem studied is
$$u_{tt}-\Delta u=|u|^{p-2}u \qquad \mbox{in
$[0,\infty)\times\Omega$,}$$
$$ u=0\qquad \mbox{on
$(0,\infty)\times\Gamma_0$,}$$
$$ \partial_\nu u
=-\alpha(x)\left(|u_t|^{m-2}u_t+\beta |u_t|^{\mu-2}u_t\right) \qquad
\mbox{on $(0,\infty)\times\Gamma_1$,}$$ $$ u(0,x)=u_0(x),\qquad
u_t(0,x)=u_1(x)\mbox{in $\Omega$,}$$ where
$\partial\Omega=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap \Gamma_1=\emptyset$, $\sigma(\Gamma_0)>0$, $21$, $\alpha\in L^\infty(\Gamma_1)$, $\alpha\ge 0$, $\beta\ge 0$. The initial data are posed in the energy space. The aim of the talk is to present some recent improvements to previous blow--up results concerning the problem. |
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