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

Abstract

We consider the nonlinear hyperbolic equation \begin{align*} u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p} \end{align*} posed in $Q:=(0,\infty )\times \mathbb{R}^{N},$ where $D_{+}^{\alpha }u$, 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.