Classification of blow-ups and monotonicity formula for half-Laplacian nonlinear heat equation
Classification of blow-ups and monotonicity formula for half-Laplacian nonlinear heat equation
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DOI：
10.1007/s00526-021-01924-8
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发表时间：
2020-02
影响因子：
2.1
通讯作者：
B. Deng;Y. Sire;Juncheng Wei;Ke Wu
中科院分区：
文献类型：
--
作者：
B. Deng;Y. Sire;Juncheng Wei;Ke Wu
考虑非线性半拉普拉斯热方程$$\开始{aligned} u_t+（-\Delta）^{\frac{1}{2}} u-|u| ^{p-1}u= 0，\quad {\mathbb {R}}^n\times（0，T）. \end{aligned}$$我们证明了所有的blow-up都是I型的，条件是且其中有一个显式指数低于临界Sobolev指数。我们证明的核心是半拉普拉斯算子的Giga-Kohn型单调性公式和自相似非线性热方程的Liouville型定理。这是在非局部方程水平上的单调性公式的第一个实例，而没有调用半空间的扩展。
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 thatandwhereis an explicit exponent which is below, 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.