A Fujita-type threshold for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces

Abstract

In this paper, we consider the semilinear damped wave equation with Hartree-type nonlinearity $\mathcal{I}_\gamma\left(|u|^{p_1}\right)|u|^{p_2}$, where $0<\gamma 0$. This formulation is particularly suited to the nonlinear analysis. We then establish the existence of a unique global mild solution for sufficiently small initial data whenever $$ p_1+p_2\geq 1+\frac{4+2\gamma}{n+2\beta}, $$ under the remaining admissibility conditions stated in the existence theorem. In particular, the critical case is covered whenever the critical line satisfies these conditions. Conversely, for initial data satisfying an explicit positive lower bound, the test-function method rules out global weak solutions when $$ 2<p_1+p_2<1+\frac{4+2\gamma}{n+2\beta}. $$ Thus, whenever the critical line is admissible and the subcritical interval is nonempty, $1+(4+2\gamma)/(n+2\beta)$ gives a Fujita-type threshold for the sum $p_1+p_2$.