Non-uniqueness for an energy-critical heat equation on $\mathbb{R}^2$

TL;DR

This paper constructs a singular solution to a nonlinear Schrödinger equation with energy-critical growth in two dimensions and demonstrates non-uniqueness of solutions for the related heat equation, highlighting the regularizing effect.

Contribution

It introduces a novel singular solution for an energy-critical nonlinear Schrödinger equation and uses it to prove non-uniqueness in the associated heat equation.

Findings

Existence of a singular solution with energy-critical nonlinearity.

Non-uniqueness of strong solutions for the semilinear heat equation.

Explicit computation showing heat equation's regularizing effect.

Abstract

We construct a singular solution of a stationary nonlinear Schr\"{o}dinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.