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


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)


We consider the nonlinear heat equation $u_t = \Delta u + {\lvert u \rvert}^{\sigma} u$ with $\sigma \gt 0$, either on $\mathbb{R}^N, N \geq 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) \sigma \lt 4$, for every $\mu \in \mathbb{R}$, if the initial value $u_0$ satisfies $u_0 (x) = \mu {\lvert x-x_0 \rvert}^{\frac{2}{\sigma}}$ in a neighborhood of some $x_0 \in \Omega$ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0) = u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $\mu {\lvert x-x_0 \rvert}^{\frac{2}{\sigma}}$ on $\mathbb{R}^N$.

Moreover, if $\mu \geq \mu_0$ for a certain $\mu_0 (N, \sigma) \geq 0$, and $u_0 \geq 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0) = u_0$, but there are infinitely many sign-changing solutions.


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