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.

中文翻译：

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具有对数非线性的热方程中的超指数增长或衰减

我们在实线上考虑具有对数非线性的热方程。对于非线性前面的一个合适的符号，我们建立了柯西问题解的存在唯一性，对于适应良好的初始数据类。高斯数据情况下的显式计算会导致各种场景，这些场景比仅与 ODE 机制进行比较更丰富，涉及（如在 ODE 情况下）长时间的双指数增长或衰减。最后，我们证明，在紧凑支持的初始数据的情况下，这种现象仍然存在。