Regularizing a nonlinear integroparabolic Fokker-Planck equation
with space-periodic solutions. Existence of strong solutions
Denis R. Akhmetov, Mikhail M. Lavrentiev, Jr., and Renato Spigler

Numerous phenomena, pertaining to Physics, Biology, Medicine, Neural Networks, are reasonably described in terms of large populations of nonlinearly coupled, often noisy, oscillators. A mathematical model is then represented by a large system of possibly stochastic nonlinear ordinary differential equations. In the limit case of infinitely many oscillators, when the interaction is of the so-called "mean-field" type, a single nonlinear parabolic model equation, containing an integral term, was derived by Kuramoto. However, improving the finite-dimensional model to take into account certain observed features, lead to the introduction of second derivatives on the left-hand side of the system above. This suggested a more general nonlinear partial differential equation, obtained by a procedure similar to that above. Such a new model equation is a Fokker-Planck type equation.

The solutions to the equation plus suitable initial data is required to be positive, periodic in angle variable, and normalized. In this paper, we address the problem of existence of such solutions.

The problem should be considered nonstandard by several reasons, namely:

• The governing equation is of the second order with respect to some variable, but only of the first order with respect to other variables. Therefore, even besides of the integral term, the equation is neither of the parabolic nor of the hyperbolic type.

• The governing equation is considered in a domain unbounded in some variable, which serves as a coefficient of the equation. This fact gives rise to singularity phenomena typical for equations with unbounded coefficients.

• The governing equation contains an integral term taking on unbounded domain.

• There is an additional variable, the natural frequency of the oscillators, with respect to which no derivatives appear, but the integral is also made with respect to it.

• We are interested only in solutions periodic in angle variable, while the governing equation contains the first time-derivative with respect to the angle variable (cf. "time-periodic solutions to parabolic equations").

Therefore, results available in the literature concerning nonlinear parabolic or even integroparabolic equations cannot be applied.

The idea is to "regularize" the equation adding a diffusion term with respect to angle variable, since the equation should be considered fully degenerate with respect to the angle variable.

Due to the results in the PhD thesis and some latest results (unpublished in the present) the existence theorem of strong solutions is proved for the original problem.