Critical exponent for semi-linear wave equations with double damping terms in exterior domains (Q2291744)

Considered is an IBVP for the damped wave equation in an exterior domain $\Omega\subset\mathbb{R}^n$, $n\geq 2$, \[ \begin{cases} u_{tt}(t,x) - \Delta u(t,x)+u_t(t,x)-\Delta u_t(t,x) = |u(t,x)|^p &\text{for }(t,x)\in (0,\infty)\times\Omega,\\ u (0, x) = u_0(x), u_t(0, x) = u_1(x), &\text{for }x\in\Omega,\\ u(t,x)=0 &\text{for }x\in\partial\Omega,\ t>0. \end{cases} \] One assumes initial data $[u_0,u_1]\in H_0^1(\Omega)\times L^2(\Omega)$. The aim is to investigate the critical exponent. One proves that for $n=2$ and $p>2$ or for $n=3,4,5$ and $1+4/(n+2) < p\leq n/(n-2)$, under additional assumptions, the mixed problem has a unique global weak solution which satisfies some decay estimates. For $n=2$, the lower bound coincides with the Fujita exponent, $p_F=1+2/n=2$. One mentions that for $n\geq 3$ the results are not optimal. Conditions under which there exists no global in time solution for the mixed problem are investigated.