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

Abstract

We construct a singular solution of a stationary nonlinear Schrodinger 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 mild 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.