TY - JOUR

T1 - Conservative parabolic problems: nondegenerated theory and degenerated examples from population dynamics

AU - Danilkina, Olga

AU - Souza, Max O.

AU - Chalub, Fabio A. C. C.

N1 - Erasmus Mundus, Grant/Award Number: Erasmus Mundus Action 2 MULTIC programme of the European Union - EMA 2 MULTIC 11-765; CNPq, Grant/Award Number: 308113/2012-8; Centro de Matematica e Aplicacoes, Universidade Nova de Lisboa, Grant/Award Numbers: UID/MAT/00297/2013 and Investigador FCT

PY - 2018/8/1

Y1 - 2018/8/1

N2 - We consider partial differential equations of drift-diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (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, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure-valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0+. The time evolution of these singular masses is also explicitly described and, as a by-product, uniqueness of this measure solution is obtained.

AB - We consider partial differential equations of drift-diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (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, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure-valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0+. The time evolution of these singular masses is also explicitly described and, as a by-product, uniqueness of this measure solution is obtained.

UR - http://www.scopus.com/inward/record.url?scp=85046028439&partnerID=8YFLogxK

U2 - 10.1002/mma.4901

DO - 10.1002/mma.4901

M3 - Article

AN - SCOPUS:85046028439

SN - 0170-4214

VL - 41

SP - 4391

EP - 4406

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

IS - 12

ER -