Diffusion processes on an interval under linear moment conditions

Abstract

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.

Well-posedness and spectral properties of heat and wave equations with non-local conditions

“…We are not aware of earlier investigations about the possibility of replacing a condition on the moment of order one by a condition on some moment of higher order. In the companion papers [25,24] we have further developed our techniques in order to address these issues.…”

On moments-preserving cosine families and semigroups in C[0, 1]

“…By (2. 19) this proves that f belongs to D(A) and we have Af = f ′′ . Finally, we check that there is a λ > 0 such that for all g ∈ C[0, 1] there exists f ∈ D such that λf − f ′′ = g. Since λ − A is injective for some large λ and its range is C[0, 1], this will show that D cannot be a proper subset of D(A) (see e.g.…”

“…A big question is of course if the requirement that two moments, say F i and F j (i = j), are preserved, determines a cosine family or a semigroup generated by a Laplace operator. A generation theorem for semigroups related to moments F 0 and F j with arbitrary j = 0 has been obtained in [19,Thm 3.4].…”

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

“…Similar restrictions also appear when considering other problems: There are counterexamples that show that not every choice of conserved moments yields a well‐posed problem for the porous media equation—cf Vázquez . However, the choice of the 0th‐order moment (mass conservation) and a further higher‐order moment may yield a well‐posed problem as shown in Mugnolo and Nicaise …”

“…The number of linear conservation laws that can be imposed, while still leading to a well‐posed problem, is also of relevance; in Mugnolo and Nicaise, a full family of moments is considered. However, well‐posedness of the problem requires that at most 2 of them are prescribed.…”