Finite time blow-up for damped wave equations with space-time dependent potential and nonlinear memory
The article refers to the Cauchy problem for the nonlinear damped wave equation \[ \begin{aligned} u_{tt}-\Delta u+a(x)b(t)u_t={J^{\delta}}_{0|t}(|u|^p)(t),\\ u(0,x)=u_o(x),\quad u_t(0,x)=u_1(x),\quad t0, x\in \mathbb{R}^n, \end{aligned} \] where $\delta=1-\gamma$ and ${J^{\delta}}_{0|t}$ is the Riemann-Liouville integral operator. The coefficient of the damping term is given by $a(x)b(t):=a_o(1+{|x|^2})^{-{\alpha}/2}(1+t)^{-\beta}$ and initial data are assumed to belong to the energy space. In cases $n=1,2$ and $p\in(1,\infty)$ or $n\geq 3$ and $1p\leq n/(n-2)$, the existence of a unique maximal mild solution is proved. In both cases, under some additional assumptions on coefficients and data one proves that the mild solution of the stated problem blows up in a finite time. For the proof, the well known test function method is applied.
scientific article; zbMATH DE number 7031110
scientific article