On the profile of energy concentration at blow-up points for sub-conformal focusing nonlinear waves

TL;DR

This paper investigates the energy distribution at blow-up points for sub-conformal focusing nonlinear wave equations, introducing new localized estimates and bounds that enhance understanding of singularity behavior.

Contribution

It derives novel bounds on energy concentration using Carleman-type inequalities, providing new insights into the blow-up dynamics of nonlinear waves.

Findings

Bounds restrict energy distribution inside cones at singularities

New localized estimates via Carleman inequalities

Applications to decay estimates for regular solutions

Abstract

We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the $H^1$-norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this $H^1$-energy can be distributed inside such cones. Our proof relies on new localized estimates obtained using Carleman- type inequalities for such nonlinear waves. These bound the $L^{p+1}$-norm in the interior of timelike cones by their $H^1$-norm near the boundary of the cones. Such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations, in the interior of time cones.