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arXiv · 2306.15234

Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation

Ryunosuke Kusaba·Tohru Ozawa
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Abstract

We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.

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Ryunosuke Kusaba, Tohru Ozawa. 2023-06-27. Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation. https://arxiv.org/abs/2306.15234

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