On Uniqueness for supercritical nonlinear wave and Schr\"odinger equations

TL;DR

This paper extends the uniqueness results for classical solutions of nonlinear wave and Schrödinger equations to include more general nonlinearities with higher growth or oscillations, improving upon previous conditions.

Contribution

It broadens the class of nonlinearities for which uniqueness of solutions can be established, including those with exponential growth.

Findings

Uniqueness holds for a wider class of nonlinearities.

Includes nonlinearities with exponential growth or oscillations.

Improves upon previous conditions for solution uniqueness.

Abstract

In a recent paper, Struwe considered the Cauchy problem for a class of nonlinear wave and Scr\"odinger equations. Under some assumptions on the nonlinearities, it was shown that uniqueness of classical solutions can be obtained in the much larger class of distribution solutions satisfying the energy inequality. As pointed out in [38], the conditions on the nonlinearities are satisfied for any polynomial growth but they fail to hold for higher growth (for example ${\rm e}^{u^2}$). Our aim here is to improve Struwe's result by showing that uniqueness holds for more general nonlinearities including higher growth or oscillations.