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

TL;DR

This paper identifies a new critical exponent for the semilinear damped wave equation with Hartree nonlinearity, establishing conditions for global existence or finite-time blow-up of solutions based on initial data in homogeneous Besov spaces.

Contribution

It introduces a novel critical exponent for the equation with Besov space initial data, expanding understanding of solution behavior in different regimes.

Findings

Global existence for supercritical and critical regimes when p_1+p_2 ≥ p_Fuji.

Finite-time blow-up for subcritical regime p_1+p_2 < p_Fuji.

Decay estimates derived for solutions with Besov space initial data.

Abstract

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