Dynamics of Partial Differential Equations

Volume 16 (2019)

Number 2

Perturbations of self-similar solutions

Pages: 151 – 183

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n2.a3

Authors

Thierry Cazenave (Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions, Paris, France)

Flávio Dickstein (Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions, Paris, France; and Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil)

Ivan Naumkin (Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma, México)

Fred B. Weissler (Université Paris 13, Sorbonne Paris Cité, LAGA CNRS UMR 7539, Villetaneuse, France)

Abstract

We consider the nonlinear heat equation ut=Δu+|u|σu with σ>0, either on RN,N1, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case (N2)σ<4, for every μR, if the initial value u0 satisfies u0(x)=μ|xx0|2σ in a neighborhood of some x0Ω and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition u(0)=u0. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value μ|xx0|2σ on RN.

Moreover, if μμ0 for a certain μ0(N,σ)0, and u00, then there is no nonnegative local solution of the heat equation with the initial condition u(0)=u0, but there are infinitely many sign-changing solutions.

Keywords

nonlinear heat equation, self-similar solutions, local existence, nonexistence, non-uniqueness

2010 Mathematics Subject Classification

Primary 35K58. Secondary 35A01, 35A02, 35C06, 35K91.

Received 14 May 2018

Published 14 March 2019