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| Fractional laplacians arise in some models of enhanced anomalous diffusion, when the Gaussian statistics of the classical Brownian motion is replaced by a different one, giving rise to the L\'evy jumps (or flights). As such operators are of real interest both in population dynamics and in relativistic quantum electrodynamics, we plan to extend the theory in this direction. We shall report about the asymptotic analysis and the study of the nodal set in case of fractional laplacians, New tools, involving different extremality conditions and new monotonicity formulas, are needed to attack this problem and raise new challenging spectral problems.
Contrary to the usual competition-diffusion cases, we observe the emergence of dramatically different phenomena depending on the type of competitive interaction. |
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