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

Slim Ibrahim; Hiroaki Kikuchi; Kenji Nakanishi; Juncheng Wei

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

Analysis of PDEs 2019-03-19 v1

Authors: Slim Ibrahim , Hiroaki Kikuchi , Kenji Nakanishi , Juncheng Wei

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.

Keywords

energy-critical heat equation non-uniqueness singular solution stationary nonlinear schrödinger equation square-exponential nonlinearity trudinger-moser inequality cauchy problem

Cite

@article{arxiv.1903.06729,
  title  = {Non-uniqueness for an energy-critical heat equation on $\mathbb{R}^2$},
  author = {Slim Ibrahim and Hiroaki Kikuchi and Kenji Nakanishi and Juncheng Wei},
  journal= {arXiv preprint arXiv:1903.06729},
  year   = {2019}
}

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

Abstract

Keywords

Cite

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