Global Solutions with Small Initial Data to Semilinear Wave Equations with Energy Supercritical Powers

arXiv2211.01594MaRDI QIDQ6416016

Chengbo Wang, Kerun Shao

Publication date: 3 November 2022

Abstract: Considering $1+n$ dimensional semilinear wave equations with energy supercritical powers $p>1+4/(n-2)$, we obtain global solutions for any initial data with small norm in $H^{s_c}imes{H}^{s_c-1}$, under the technical smooth condition $p>s_c-s_0$, with certain $s_{0}le1$ and $s_c=n/2-2/(p-1)$. When $pgeq2$, we could take $s_0=1$. In particular, combined with previous works, our results give a complete verification of the Strauss conjecture, up to space dimension $9$. The higher dimensional case, $nge10$, seems to be unreachable, in view of the wellposed theory in $H^s$.

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