Energy scattering for 2D critical wave equation

TL;DR

This paper establishes the existence and asymptotic behavior of wave operators for 2D critical nonlinear Klein-Gordon and Schrödinger equations with exponential nonlinearity, using concentration analysis and Bourgain's induction.

Contribution

It introduces a novel approach to handle the critical case by tracking energy concentration, extending wave operator results to exponential nonlinearities in two dimensions.

Findings

Solutions approach free solutions at infinity below or at critical energy.

Concentration of energy affects nonlinear estimates in the critical case.

Method applies to both Klein-Gordon and Schrödinger equations with exponential nonlinearity.

Abstract

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation.