Blow-up of the critical norm for a supercritical semilinear heat equation

Hideyuki Miura; Jin Takahashi

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arXiv · 2206.10790

Blow-up of the critical norm for a supercritical semilinear heat equation

Hideyuki Miura·Jin Takahashi

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Abstract

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range $p>p_S$ is optimal in view of the existence of type II blow-up solutions with bounded critical norm for $p=p_S$.

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Hideyuki Miura, Jin Takahashi. 2022-06-22. Blow-up of the critical norm for a supercritical semilinear heat equation. https://arxiv.org/abs/2206.10790

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