Nonexistence of global solutions to semi-linear fractional evolution equation

Abstract

In this paper, we consider the following semi-linear fractional evolution equation
$$ u_{tt}+(-\Delta)^{\frac{\beta}{2}}u+D^{\alpha}_{0\mid t}u=h(t,x)\left|u\right|^{p}, $$

posed in $(0,T)\times \mathbb{R}^{N},$ where
$(-\Delta)^{\frac{\beta}{2}},\ 0<\beta \leq 2$ is
$\frac{\beta}{2}-$ fractional power of $-\Delta$, and $D^{\alpha}_{0/t}$ denotes the derivatives of order $\alpha$ in the sense of Caputo. The nonexistence of global solutions theorem is established. Our method of proof is based on suitable choices of the test functions in the weak
formulation of the sought solutions.