A note on the critical range for some inhomogeneous equations with a power nonlinearity

In this paper, we consider the initial value problem for a nonhomo-geneous second-order semilinear equation with fractional powers of the Laplace operator and with a power nonlinearity. The model considered includes the Boussinesq equation. We obtain nonexistence of global-in-time solutions for subcritical power nonlinearities, where the critical exponent is in the sense of Lee-Ni. We also obtain the nonexistence of local solutions for supercritical power nonlinearities with an adapted version of the Lee-Ni critical exponent. We also provide a result of global existence of solutions with small initial energy that is valid in the range between the two previous critical exponents, provided that a blow-up alternative is first proved for local solutions.