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: u(tt) - Delta u + (-Delta)(beta/2) D-+(alpha) u = vertical bar u vertical bar(p) where (-Delta)(beta/2), 1 <= beta <= 2 stands for the beta/2 fractional power of the Laplacien and D-+(alpha) 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. (c) 2004 Published by Elsevier Inc.