Geodesic
Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations
Electronic journal of differential equations, Tome 2002 (2002)
A nonlinear Fokker-Planck type ultraparabolic integro-differential equation is studied. It arises from the statistical description of the dynamical behavior of populations of infinitely many (nonlinearly coupled) random oscillators subject to "mean-field" interaction. A regularized parabolic equation with bounded coefficients is first considered, where a small spatial diffusion is incorporated in the model equation and the unbounded coefficients of the original equation are replaced by a special "bounding" function. Estimates, uniform in the regularization parameters, allow passing to the limit, which identifies a classical solution to the original problem. Existence and uniqueness of classical solutions are then established in a special class of functions decaying in the velocity variable.
Classification : 35K20, 35K60, 45K05
Keywords: nonlinear integro-differential parabolic equations, ultraparabolic equations, Fokker-Planck equation, degenerate parabolic equations, regularization 
@article{EJDE_2002__2002__a156,
     author = {Akhmetov,  Denis R. and Lavrentiev,  Mikhail M.jun. and Spigler,  Renato},
     title = {Existence and uniqueness of classical solutions to certain nonlinear integro-differential {Fokker-Planck} type equations},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {1011.35068},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a156/}
}
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AU  - Akhmetov,  Denis R.
AU  - Lavrentiev,  Mikhail M.jun.
AU  - Spigler,  Renato
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JO  - Electronic journal of differential equations
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VL  - 2002
UR  - http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a156/
LA  - en
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%A Spigler,  Renato
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%J Electronic journal of differential equations
%D 2002
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%U http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a156/
%G en
%F EJDE_2002__2002__a156
Akhmetov,  Denis R.; Lavrentiev,  Mikhail M.jun.; Spigler,  Renato. Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a156/