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 $0<\gamma <n$, with initial data possessing additional negative regularity in ${\dot{B}}_{2,\infty }^{-\beta }$, $\beta \in [0,n/2)$. We first establish decay estimates for the corresponding linear equation in homogeneous Besov spaces ${\dot{B}}_{q,r}^s$. A key point is that the low-frequency part of the data is measured only in ${\dot{B}}_{m,\infty }^{-\beta }$, while the heat-like behavior yields the required ${\ell }^r$ summability under the condition $s+\beta +n(1/m-1/q)>0$. 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 $$p_1+p_2\ge 1+\frac{4+2\gamma }{n+2\beta },$$ 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 $p_1+p_2$.

23 pages