The application of decay character on the global behavior of damped wave equation with Riesz potential-type power nonlinearity

In this paper, our first objective is to investigate the decay rates and the global (in time) existence of solutions to the semilinear damped wave equation with the Riesz potential-type power nonlinearity $\mathcal{I}_\gamma\left(|u|^p\right)$, where $\gamma\in[0,n)$, in terms of the decay character of the initial data.

This approach enables us to establish global existence results for several classes of initial data.

Our second objective is to show, via a blow-up argument, that the conditions imposed on the nonlinearity in the global existence theorem are sharp for initial data belonging to the pseudo-measure space $\mathcal{Y}^q$.

As a consequence, we derive the new critical exponent $$ p_{\mathrm{crit}}(n,q,\gamma):=1+\frac{2+\gamma}{n-q} $$ for $1\leq n\leq 4$ and $0\leq \gamma<q<n/2$.

Furthermore, we establish a sharp lifespan estimate for solutions that blow up in finite time.