Existence and Uniqueness of Classical Solutions to Certain Nonlinear Integro-Differential Fokker-Planck Type Equations




Akhmetov, Denis R.
Lavrentiev, Mikhail M.
Spigler, Renato

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Southwest Texas State University, Department of Mathematics


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.



Nonlinear integro-differential parabolic equations, Ultraparabolic equations, Fokker-Planck equation, Degenerate parabolic equations, Regularization


Akhmetov, D. R., Lavrentiev, M. M., & Spigler, R. (2002). Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations. <i>Electronic Journal of Differential Equations, 2002</i>(24), pp. 1-17.


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