The critical exponent for the two-dimensional semilinear wave equation with strong damping

Abstract

In this manuscript, we determine the critical exponent for the two-dimensional semilinear wave equation with strong damping. A recent result in D&#39;Abbicco (arXiv, 2026) gives global in-time small data solutions for $p>\frac{10}{3}$, whereas we in the paper prove finite-time blow-up for $2<p\leqslant\frac{10}{3}$ under an averaged sign condition on the initial velocity. Together with the previously known blow-up result for $1<p\leqslant3$, the threshold for the power nonlinearity $|u|^p$ is \begin{align*} p=p_{\mathrm{crit}}=\frac{10}{3}. \end{align*} We prove positivity of the full two-dimensional velocity fundamental solution and construct a positive averaged kernel, which allows us to establish a nonlinear lower-bound iteration in a parabolic region.