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 with , either on , or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case , for every , if the initial value satisfies in a neighborhood of some and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition . The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value on .
Moreover, if for a certain , and , then there is no nonnegative local solution of the heat equation with the initial condition , 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