Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equation...

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators [8, 9, 6]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scaleinvariant Harnack inequality fo...

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be C.

Abstract We study the singular set in thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ |y|a $$a \in (-1,1)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">a<mml:...

A model for phase separation in multicomponent systems is studied. In particular the possibility of a concentration dependence of the mobility matrix is taken into account. This leads to a system of fourth order degenerate parabolic partial diierential equations. We derive these equations from balance laws, show some properties of the model and prove a global existence result for the degenerate...