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

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.