I want to know how to build the fast decaying solution for
(-\Delta)^\alpha u=u^p in R^N\setminus {0}
where p>\frac{N}{N-2\alpha}.
Here fast decay means the solution hjas decaying rate |x|^{2\alpha-N}.
For \alpha=1, Dynamic method was applied. But for fractional laplacian, how
can we build the fast decaying singuar solutions?
By Paley-Wiener type arguments the solutions cannot decay too fast....
Very surprised! I will try to understand the Paley-Wiener type arguments. Thanks!
May you give the address or link of paper or article (including this equation) so I can see the discussion about it, I would make sure I got your meaning truly and then perhaps I can give you some idea for finding the numerical solution.
This is due to the fact that the linear symbol for fractionary equations is not smooth and this prevents exponentail decay. See for instance the decay properties of solitary waves to the Benjamin-Ono or KP I equations. See the attached papers.
@jean-Claude Saut Thanks for these articles,
that is true, those are difficult cases, I am working on the numerical solution of RLW now (one dimension, and not fractional order) which is easier than these cases, but still the numerical methods are not too fast and I am trying to use more techniques to construct the approximated solution with more accuracy. The results may be useful for fractional case or other PDEs as KP in future.
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