A new critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces

In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form ${\mathcal{I}}_{\gamma }\left(\mathrm{\mid }u{\mathrm{\mid }}^{p_1}\right)\mathrm{\mid }u{\mathrm{\mid }}^{p_2},p_1,p_2>0,\gamma \in [0,n)$, with initial data taken in the homogeneous Besov spaces ${\dot{B}}_{2,\mathrm{\infty }}^{-\beta }$, where $\beta \in [0,\frac{n}{2})$. Our approach is based on deriving decay estimates for solutions to the associated linear damped wave equation with initial data belonging to ${\dot{B}}_{2,\mathrm{\infty }}^{-\beta }$, combined with refined tools from Harmonic Analysis. As a consequence, we identify a new critical exponent given by $$p_1+p_2:=p_{\mathrm{F}\mathrm{u}\mathrm{j}\mathrm{i}}\left(\frac{n+2\beta }{2+\gamma }\right):=1+\frac{4+2\gamma }{n+2\beta }\text{ for }\beta \in [0,\frac{n}{2})\text{ and }\gamma \in [0,n)\mathrm{.}$$ More precisely, we establish the global (in time) existence of small data solutions in the supercritical and critical regimes $p_1+p_2\ge p_{\mathrm{F}\mathrm{u}\mathrm{j}\mathrm{i}}\left(\frac{n+2\beta }{2+\gamma }\right)$. In contrast, we prove finite-time blow-up of weak solutions, even for arbitrarily small initial data, in the subcritical range $2<p_1+p_2<p_{\mathrm{F}\mathrm{u}\mathrm{j}\mathrm{i}}\left(\frac{n+2\beta }{2+\gamma }\right)$.