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

TL;DR

This paper investigates both uniform and degenerated parabolic PDEs with conservation laws, focusing on population dynamics models with absorbing boundaries, and introduces a regularisation approach for measure solutions in degenerated cases.

Contribution

It develops a unified framework for analyzing degenerated and non-degenerated parabolic PDEs with conservation laws, especially in population dynamics contexts.

Findings

Uniform parabolic problems are fully characterized.

Degenerated problems are handled via regularisation, ensuring unique measure solutions.

Two population dynamics examples illustrate the theoretical results.

Abstract

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.