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.
Categories: math.AP