Diffusion processes on an interval under linear moment conditions

TL;DR

This paper explores diffusion equations on a bounded interval where boundary conditions are replaced by linear moment conditions, analyzing well-posedness and long-term behavior including decay rates for various nonlinear cases.

Contribution

It introduces a novel approach of replacing boundary conditions with linear moment conditions and studies their effects on well-posedness and asymptotic decay in diffusion equations.

Findings

Well-posedness established for heat equations with moment conditions.

Polynomial decay in porous medium range.

Exponential decay in fast diffusion range.

Abstract

We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of another arbitrarily chosen order n. Each choice of n induces the addition of a certain potential in the equation, the case of zero potential arising exactly in the special case of n=1 corresponding to a condition on the barycenter. In the linear case we exploit smoothing properties and perturbation theory of analytic semigroups to obtain well-posedness for the classical heat equation (with said conditions on the moments). Long time behavior is studied for both the linear heat equation with potential and certain nonlinear equations of porous medium or fast diffusion type. In particular, we prove polynomial decay in the porous medium range and exponential decay in the fast diffusion range, respectively.