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

Abstract

We consider the one-dimensional heat and wave equations but -instead of boundary conditionswe impose on the solution certain non-local, integral constraints. An appropriate Hilbert setting leads to an integration-by-parts formula in Sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well-posedness, hence sharpening regularity results previously obtained by other authors. In doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions, like periodic ones. In the self-adjoint case, we even obtain eigenvalue asymptotics of so-called Weyl's type.2000 Mathematics Subject Classification. 47D06, 35J20, 34B10.

Waves and diffusion on metric graphs with general vertex conditions

“…6.5], see Subsection 3.6. We can also treat different non-local boundary conditions (for example those studied in [37], see Example 2.10). This paper is organized as follows.…”

Diffusion processes on an interval under linear moment conditions

“…This special case had already been discussed by A. Bouziani and his coauthors starting with [BB96], see [MN11] for more detailed references. A somehow comparable approach has been followed in [Váz07,§ 9.6], where an analysis based on mere finiteness of mass is performed.…”

“…Beginning with [Can63], many authors have studied linear partial differential equations equipped with conditions on the moments complementing those on the boundary values. In [MN11] the present authors have gone on to observe that, in fact, in the case of the linear heat equation one can drop the boundary conditions and obtain well-posedness under a wide class of linear conditions on the moments of order 0 and 1 -and in particular, whenever both of them are assumed to vanish constantly, i.e., (1.2) µ n (u(t)) = 0 ∀t ≥ 0, n = 0, 1.…”

A general theorem on generation of moments-preserving cosine families by Laplace operators in C[0,1]

“…Before continuing, we note that in case (a), a generation theorem for momentspreserving semigroups has been obtained in [9,Theorem 3.4].…”

“…Following the seminal work of J.R. Cannon [6], a semigroup-theoretical study of diffusion and wave equations associated with one-dimensional Laplace operators equipped with integral conditions has recently been commenced in [9], where an abstract framework for studying such problems in Hilbert spaces has been proposed. Paper [5] presents a different approach, applicable apparently in a broader context: it shows that the recently developed Lord Kelvin's method of images [3,4] provides natural tools for constructing moments-preserving cosine families.…”