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

## TL;DR

This paper studies the heat equation with a logarithmic nonlinearity, establishing solution existence and uniqueness, and reveals complex growth and decay behaviors that extend to compactly supported initial data.

## Contribution

It provides the first rigorous analysis of superexponential growth and decay phenomena in the heat equation with logarithmic nonlinearities.

## Key findings

- Existence and uniqueness of solutions for certain initial data
- Explicit solutions for Gaussian initial data show rich dynamics
- Superexponential growth or decay persists for compactly supported data

## Abstract

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data.