论文标题
半拉普拉斯非线性热方程的爆炸和单调性公式的分类
Classification of Blow-ups and Monotonicity Formula for Half Laplacian Nonlinear Heat Equation
论文作者
论文摘要
我们认为非线性半拉普拉斯热方程$$ u_t+（ - δ）^{\ frac {1} {2}} u- | u | u |^{p-1} u = 0，\ quad \ quad \ mathbb {r}^n \ times（0，t）。 $$我们证明所有吹动都是I类型，但前提是$ n \ leq 4 $和$ 1 <p <p <p <p _ {*}（n）$其中$ p _ {*}（n）$是一个明确的指数，它低于$ \ frac {n+1} {n-1} {n-1} $，即关键的sobolev exponent。我们证明的核心是半laplacian的Giga-kohn型单调公式和一种用于自相似非线性热方程的Liouville型定理。这是非本地方程级别的单调性公式的第一个实例，而无需调用半空间的扩展。
We consider the nonlinear half laplacian heat equation $$ u_t+(-Δ)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T). $$ We prove that all blows-up are type I, provided that $n \leq 4$ 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.
