On the semilinear damped wave equation with Riesz potential-type power nonlinearity and initial data in pseudo-measure spaces

TL;DR

This paper determines the critical exponent for a damped wave equation with Riesz potential nonlinearity and initial data in pseudo-measure spaces, establishing conditions for global existence or finite-time blow-up of solutions.

Contribution

It introduces a new critical exponent for the equation and proves global existence or blow-up results based on this exponent, using decay estimates and harmonic analysis tools.

Findings

Derived the critical exponent p_crit(n, q, γ) = 1 + (2 + γ)/(n - q).

Proved global existence of small solutions for p ≥ p_crit.

Established finite-time blow-up for p < p_crit.

Abstract

In this paper, our main objective is to determine the critical exponent for the semilinear damped wave equation with Riesz potential-type power nonlinearity $\mathcal{I}_\gamma(|u|^p)$ for $\gamma\geq 0$, and initial data belonging to the pseudo-measure spaces $\mathcal{Y}^q$. Our main approach is to establish decay estimates for solutions to the corresponding linear problem in the $L^2$-framework with initial data belonging to $\mathcal{Y}^q$, combined with some tools from Harmonic Analysis. Consequently, we derive a new critical exponent $$p_{\mathrm{crit}}(n, q, \gamma):=1+\frac{2+\gamma}{n-q},$$ for $1 \leq n \leq 4$ and $0 \leq \gamma < q < n/2$, by proving the global (in time) existence of small data solutions when $p \geq p_{\mathrm{crit}}(n, q, \gamma)$, and blow-up of weak solutions in finite time, even for small initial data, whenever $1<p<p_{\mathrm{crit}}(n, q, \gamma)$. Moreover, we are going to provide sharp estimates for the lifespan of solutions when a blow-up phenomenon occurs.