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

TL;DR

This paper proves the existence of global solutions for small initial data in energy supercritical semilinear wave equations up to dimension 9, confirming the Strauss conjecture in these cases.

Contribution

It establishes global solutions for small initial data in supercritical regimes, extending the Strauss conjecture verification up to nine spatial dimensions.

Findings

Global solutions exist for small data in dimensions up to 9.

Complete verification of the Strauss conjecture for these dimensions.

Limitations identified for dimensions 10 and above.

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 $H^{s_c}\times H^{s_c-1}$, under the technical smooth condition $p>s_c-\bar{s}_0$, with $\bar{s}_0= 1/2+(n-3)/(2\max(n-1-p,n-3))$ and $s_c=n/2-2/(p-1)$. 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, $n\ge 10$, seems to be unreachable, in view of the wellposed theory in $H^s$.