Critical thresholds for semilinear damped wave equations with Riesz potential power nonlinearity and initial data in pseudo-measure spaces
Tang Trung Loc, Duong Dinh Van, Phan Duc An
Abstract
In this paper, our primary objective is to establish the time decay properties of solutions $u(t, x)$ with initial data $u_0, u_1 \in \mathcal{Y}^q$ (pseudo-measure spaces) to the linear damped wave equation in the spaces $L^2\left(\mathbb{R}^n\right)$ and $\dot{H}^s\left(\mathbb{R}^n\right)$ (for $s \leq 0$ or $s \geq 0$). Our subsequent aim is to investigate the semilinear damped wave equation with a Riesz potential-type power nonlinearity $\mathcal{I}_γ\left(|u|^p\right)$, where $γ\in [0, n)$, and initial data belonging to pseudo-measure spaces $\mathcal{Y}^q$. In addition, we derive a new critical exponent $p=p_{\mathrm{crit}}(n, q, γ):=1+\frac{2+γ}{n-q}$ for some $q \in\left(0, \frac{n}{2}\right)$ and in low spatial dimensions, within the framework of pseudo-measure spaces $\mathcal{Y}^q$. Specifically, we prove the global (in time) existence of small-data Sobolev solutions with low regularity when $p \geq p_{\mathrm{crit}}(n, q, γ)$, and the finite-time blow-up of weak solutions, even for small initial data, whenever $1< p<p_{\mathrm{crit}}(n, q, γ)$. Moreover, in order to characterize the blow-up time more precisely, we establish sharp upper and lower bound estimates for the lifespan of solutions in the subcritical regime.