Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 4

Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

Pages: 343 – 358

DOI: https://dx.doi.org/10.4310/DPDE.2017.v14.n4.a2

Authors

Matthieu Alfaro (Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France)

Rémi Carles (Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France)

Abstract

We consider the heat equation with a logarithmic nonlinearity, on the real line. For a suitable sign in front of the nonlinearity, we establish the existence and uniqueness of solutions of the Cauchy problem, for a well-adapted class of initial data. Explicit computations in the case of Gaussian data lead to various scenarii which are richer than the mere comparison with the ODE mechanism, involving (like in the ODE case) double exponential growth or decay for large time. Finally, we prove that such phenomena remain, in the case of compactly supported initial data.

Keywords

heat equation, logarithmic nonlinearity, large time behavior

2010 Mathematics Subject Classification

Primary 35K05, 35K55. Secondary 35B30, 35B40, 35B51.

Full Text (PDF format)

Received 29 March 2017

Published 14 December 2017