Blow-up of the critical norm for a supercritical semilinear heat equation
Blow-up of the critical norm for a supercritical semilinear heat equation
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发表时间：
2022-06
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通讯作者：
H. Miura;J. Takahashi
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作者：
H. Miura;J. Takahashi
我们考虑了半线性热方程$u_t=\Delta u+|u|^{p-1}u$在$\mathbf{R}^n$的任意光滑域上爆破解的标度临界Lebesgue范数。在$p>p_S:=(n+2)/(n-2)$的范围内，我们证明了临界范数在爆破时间附近必须是无界的，其中不施加I型爆破条件。考虑到$p=p_S$具有有界临界范数的II型爆破解的存在，范围$p>p_S$是最优的。
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 $p=p_S$.