Solution blow‐up for a fractional in time acoustic wave equation

Abstract

We consider the fractional in time acoustic wave equation where 1<α<2, is the Caputo fractional derivative of order α, u=u(t,x), t>0, , is the pressure in the medium, ε is the nonlinear acoustic parameter, ρ0 is the equilibrium density in the medium, and c0 is the equilibrium sound velocity. We study a Cauchy problem for this equation and a mixed boundary value problem in a bounded domain. For each problem, sufficient conditions for the blow‐up of solutions are derived. Moreover, we provide a class of initial data for which there are no classical solutions even locally in time. Our approach is based on the nonlinear capacity method.