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Classification of blow-ups and monotonicity formula for half-Laplacian nonlinear heat equation

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Abstract

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.

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Acknowledgements

The research of J. Wei is partially supported by NSERC of Canada. The research of B. Deng and K. Wu is supported by China Scholar Council. The research of B. Deng is also supported by Natural Science Foundation of China (No. 1172110 and No. 11971137). We would like to thank D. Gomez for some technical support for numerical computation. We also thank H. Zaag for pointing out a mistaken statement.

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Correspondence to Yannick Sire.

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Appendix: computation of \(c_1M_1\)

Appendix: computation of \(c_1M_1\)

For \(n=1\), \(f_j(y), j=1,2\) defined by (3.11) has an explicit expression. Indeed, recall that \(\rho (y)=\frac{1}{1+y^2}\), then

$$\begin{aligned} \begin{aligned} f_1(y):=&\ \frac{1}{\rho (y)} \int _{ B_{|y|}(0)}\frac{(\rho (y')-\rho (y))^2}{|y'-y|^{2}}\frac{1}{\rho (y')}dy'\\ =&\ \frac{1}{1+y^2}\int _{ B_{|y|}(0)}\frac{(y'+y)^2}{1+y'^2}dy'\\ =&\ \frac{2}{1+y^2}\int _0^{|y|}dy' +\frac{2(y^2-1)}{1+y^2}\int _0^{|y|}\frac{dy'}{1+y'^2}\\ =&\ \frac{2|y|}{1+y^2}+\frac{2(y^2-1)}{1+y^2}\arctan |y|. \end{aligned} \end{aligned}$$
(7.1)

Similarly,

$$\begin{aligned} \begin{aligned} f_2(y):=&\ \frac{1}{\rho (y)^2} \int _{{\mathbb {R}}\setminus B_{|y|}(0)}\frac{(\rho (y')-\rho (y))^2}{|y'-y|^{2}}dy'\\ =&\ \int _{{\mathbb {R}}\setminus B_{|y|}(0)}\frac{(y'+y)^2}{(1+y'^2)^2}dy'\\ =&\ 2\int _{|y|}^\infty \frac{dy'}{1+y'^2} +2(y^2-1)\int _{|y|}^\infty \frac{dy'}{(1+y'^2)^2}\\ =&\ \pi -2\arctan |y| +(y^2-1)\big (\frac{\pi }{2}-\arctan |y| -\frac{|y|}{1+y^2}\big ). \end{aligned} \end{aligned}$$
(7.2)

Since \(c_1=\frac{1}{\pi }\), we are going to prove that \(M_1<4\pi \). Since \(f_j(y), j=1,2\) are even, we may assume \(y\in [0,+\infty )\). It is not hard to see that

$$\begin{aligned} \begin{aligned} \frac{2|y|}{1+y^2}\le&\ 1,\\ \frac{2(y^2-1)}{1+y^2}\arctan |y|\le&\ \pi ,\\ \pi -2\arctan |y|\le&\ \pi . \end{aligned} \end{aligned}$$
(7.3)

Let

$$\begin{aligned} f(y)=(y^2-1)\big (\frac{\pi }{2}-\arctan |y|-\frac{|y|}{1+y^2}\big ), \end{aligned}$$
(7.4)

then

$$\begin{aligned} f'(y)=2y\big (\frac{\pi }{2}-\arctan |y|-\frac{|y|}{1+y^2}\big ) -\frac{2(y^2-1)}{(1+y^2)^2}. \end{aligned}$$
(7.5)

Observe

$$\begin{aligned} f(y)<0=f(1),\ \text {for all}\ y\in [0,1), \end{aligned}$$
(7.6)

and, by the L’Hôpital’s rule,

$$\begin{aligned} \begin{aligned} \lim _{y\rightarrow \infty }f(y)=&\ \lim _{y\rightarrow \infty } \frac{\frac{2}{(1+y^2)^2}}{\frac{2y}{(y^2-1)^2}}\\ =&\ \lim _{y\rightarrow \infty }\frac{2}{y}=0. \end{aligned} \end{aligned}$$
(7.7)

Then f achieves its maximum at some critical point \(y_1\ge 1\). \(f'(y_1)=0\) implies that

$$\begin{aligned} \frac{\pi }{2}-\arctan y_1-\frac{y_1}{1+y_1^2}=\frac{y_1^2-1}{y_1(1+y_1^2)^2}. \end{aligned}$$
(7.8)

It follows that

$$\begin{aligned} f(y_1)=\frac{(y_1^2-1)^2}{y_1(1+y_1^2)^2}<\frac{1}{y_1}\le 1. \end{aligned}$$
(7.9)

Therefore, we conclude

$$\begin{aligned} M_1\le 1+\pi +\pi +1<4\pi . \end{aligned}$$
(7.10)

In fact, a numerical calculation shows that \(M_1\approx 4.8271<4\pi \). As a consequence, \(p_*(1)\approx 4.2072\).

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Deng, B., Sire, Y., Wei, J. et al. Classification of blow-ups and monotonicity formula for half-Laplacian nonlinear heat equation. Calc. Var. 60, 52 (2021). https://doi.org/10.1007/s00526-021-01924-8

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