Non-existence of Global Solutions for a Fractional Wave-diffusion Equations

Abstract

We considered the Cauchy problem for the fractional wave-diffusion equation $$D^αu-Δ|u|^{m-1}u+(-Δ)^{β/2}D^γ|u|^{l-1}u=h(x,t)|u|^p+f(x,t)$$ with given initial data and where p > 1, 1 < α < 2, 0 < β < 2, 0 < γ < 1. Nonexistence results and necessary conditions for global existence are established by means of the test function method. This results extend previous works.