Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping

Abstract

We consider the nonlinear hyperbolic equation

$u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x){\left|u\right|}^p$

posed in $Q:=(0,\infty )\times {\mathbb{R}}^N,$ where $D_{+}^{\alpha }u$, $0<\alpha <1$ is a time fractional derivative, with given initial position and velocity $u(0,x)=u_0(x)$ and $u_t(0,x)=u_1(x).$ We find the Fujita's exponent which separates in terms of $p,\alpha$ and $N,$ the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on $u_1(x)$ and $h(x,t)$ assuring non-existence of local solutions.