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

Abstract

We construct a singular solution of a stationary nonlinear Schrödinger 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.

Cited by in corpus (1)

• Global rough solution for $L^2$-critical semilinear heat equation in the negative Sobolev space