Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite time blow up
This paper deals with the blow-up problem for equation \[ u_{tt}-\Delta u-\omega\Delta u_t+\mu u_t = |u|^{p-2}u, \] where $(x,t)\in\Omega\times\mathbb{R}_+^*$, with Dirichlet boundary conditions, with $\omega\geq 0$, and $\mu\geq 0$. Here, $\Omega$ is a smooth bounded domain of $\mathbb{R}^n$ ($n\geq 1$), and $p>2$ if $n\leq 2$, $p\in(2, 2n/(n-2)]$ if $n\geq 3$ and $\omega>0$, and $p\in(2, 2(n-1)/(n-2)]$ if $n\geq 3$ and $\omega=0$. The main result states that solutions blow-up in finite time if the $L^2$-scalar product of the Cauchy data $u_0$ and $u_1$ is large enough for $t=0$ (see eq.~(9) of the paper). The paper relies on a paper by \textit{F. Gazzola} and \textit{M. Squassina} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 2, 185--207 (2006; Zbl 1094.35082)] on this equation.
singular perturbation of the wave equation
scientific article; zbMATH DE number 7040494
scientific article