Generalized solutions of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes
The main purpose of this paper is to construct both strong and weak solutions of the Cauchy problem for a class of systems of nonlinear parabolic equations. The authors give a stochastic interpretation of such system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, the authors use some results from Kunita theory of stochastic flows applied to the case of systems of parabolic equations. Firstly the authors develop a stochastic approach to the construction of weak solution to the Cauchy problem for a system of linear parabolic equations and then extend these results to the system of nonlinear parabolic equations.
scientific article; zbMATH DE number 6042627