Abstract
We consider the nonlinear half-Laplacian heat equation
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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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
Similarly,
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
Let
then
Observe
and, by the L’Hôpital’s rule,
Then f achieves its maximum at some critical point \(y_1\ge 1\). \(f'(y_1)=0\) implies that
It follows that
Therefore, we conclude
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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DOI: https://doi.org/10.1007/s00526-021-01924-8