Nonexistence of global solutions to a hyperbolic equation with a space-time fractional damping

We establish conditions that ensure the absence of global solutions to the nonlinear hyperbolic equation with a time-space fractional damping:utt-Δu+(-Δ)β/2D+αu=|u|p,where (−Δ)β/2, 1 ⩽ β ⩽ 2 stands for the β/2 fractional power of the Laplacien and D+α is the Riemann-Liouville's time fractional derivative [10]. Our results include nonexistence results as well as necessary conditions for the local and global solvability. The method used is based on a duality argument with an appropriate choice of the test function and a scaling argument.