A non-linear modification of the class of Kolmogorov-equations is given by

\[u_t=u_{xx}-u u_y-f \]

Here, \(u=u(x,y,t) \). The equation can be solved as a Cauchy-problem (that is, a pure initial value problem).

Referanse: [1] A. Pascucci, and S. Polidoro, On the Cauchy problem for a nonlinear Kolmogorov equation, SIAM J Math Anal, 35 (2003), 579-595

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