We consider the nonlinear half-Laplacian heat equation

$$\begin{aligned} u_t+(-\Delta )^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad {\mathbb {R}}^n\times (0,T). \end{aligned}$$

We prove that all blows-up are type I, provided that \(n \le 4\) and \( 1<p<p_{*} (n)\) where \( p_{*} (n)\) is an explicit exponent which is below \(\frac{n+1}{n-1}\), the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half-Laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension to the half-space.

中文翻译：

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半拉普拉斯非线性热方程的爆破分类和单调公式

我们考虑非线性半拉普拉斯热方程

$$ \ begin {aligned} u_t +（-\ Delta）^ {\ frac {1} {2}} u- | u | ^ {p-1} u = 0，\ quad {\ mathbb {R}} ^ n \ times（0，T）。\ end {aligned} $$

我们证明所有爆炸都是I型，前提是\（n \ le 4 \）和\（1 <p <p _ {*}（n）\）其中\（p _ {*}（n）\）为低于临界Sobolev指数\（\ frac {n + 1} {n-1} \）的显式指数。证明的中心是半拉普拉斯算子的Giga-Kohn型单调公式和自相似非线性热方程的Liouville型定理。这是在非局部方程级别上单调性公式的第一个实例，而无需调用对半空间的扩展。