Conservative parabolic problems: non-degenerated theory and degenerated   examples from population dynamics
  We consider partial differential equations (PDE) of drift-diffusion type in the unit interval, supplemented by either two conservation laws or by a conservation law and a further boundary condition. We treat two different cases: (i) uniform parabolic problems; (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, bring new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either one or two absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit that is unique. Two prototypical problems from population dynamics are treated in detail. 
Olga Danilkina and Max O. Souza and Fabio A. C. C. Chalub
math.AP