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

We consider the fractional in time acoustic wave equation 1c02∂0|tαu−Δu=εc04ρ0∂0|tαu2, where 10, x∈R3 , 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.