Dynamics for the energy critical nonlinear wave equation in high dimensions

TL;DR

This paper extends the classification of solutions near the ground state for the energy critical nonlinear wave equation to all dimensions greater than or equal to six, overcoming challenges posed by non-Lipschitz nonlinearities in high dimensions.

Contribution

It generalizes previous results to higher dimensions by utilizing decay properties of the ground state to handle non-Lipschitz nonlinearities.

Findings

Classification of solutions in dimensions d≥6

Handling of non-Lipschitz nonlinearities in high dimensions

Extension of variational analysis near ground state

Abstract

In the work by T. Duyckaerts and F. Merle, they studied the variational structure near the ground state solution $W$ of the energy critical wave equation and classified the solutions with the threshold energy $E(W,0)$ in dimensions $d=3,4,5$. In this paper, we extend the results to all dimensions $d\ge 6$. The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of $W$.