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Nonlocal quadratic evolution problems

100%

Biler P. , Woyczyński W. A.

Banach Center Publications

2000

tom 52

nr 1

11-24

Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data. Monte Carlo approximation schemes by interacting particle systems are also mentioned.

Interacting particle approximation for nonlocal quadratic evolution problems

88%

Biler P. , Funaki T. , Woyczynski W. A.

Probability and Mathematical Statistics

1999

tom Vol. 19, Fasc. 2

267--286

The existence of McKean's nonlinear jump Markov processes and related Monte Carlo type approximation schemes by interacting particle systems (propagation of chaos) are studied for a class of multidimensional doubly nonlocal evolution problems with a fractional power of the Laplacian and a quadratic nonlinearity involving an integral operator. Asymptotically, these equations model the evolution of density of mutually interacting particles with anomalous (fractal) Lévy diffusion.