Abstract
We consider the fractional in time acoustic wave equation
1/c(0)(2) partial derivative(alpha)(0 vertical bar t) u - Delta u = epsilon/c(0)(4)rho(0) partial derivative(alpha)(0 vertical bar t)u(2),
where 1<alpha<2, partial derivative(alpha)(0 vertical bar t) is the Caputo fractional derivative of order alpha, u=u(t,x), t>0, x is an element of R3, is the pressure in the medium, epsilon is the nonlinear acoustic parameter, rho(0) is the equilibrium density in the medium, and c(0) 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.