Diffusion processes on an interval under linear moment conditions

Delio Mugnolo , Serge Nicaise (1)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

We discuss a class of linear and nonlinear 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.

Mots clés: Nonlocal conditions for PDEs; Porous medium equation; Heat equation; Subdifferentials

Domaines: Mathématiques [math]

Delio Mugnolo, Serge Nicaise. Diffusion processes on an interval under linear moment conditions. Semigroup Forum, 2014, 88 (2), pp.479-511. ⟨10.1007/s00233-013-9552-1⟩. ⟨hal-03163692⟩