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