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

TL;DR

This paper studies heat and wave equations with non-local integral conditions, establishing well-posedness, regularity, and spectral properties using semigroup theory and Sobolev space analysis.

Contribution

It introduces a unified framework for non-local conditions, extends regularity results, and derives eigenvalue asymptotics, connecting to classical boundary conditions.

Findings

Proves analytic well-posedness of equations with non-local conditions.

Provides eigenvalue asymptotics of Weyl's type for self-adjoint cases.

Connects non-local integral conditions with traditional boundary conditions.

Abstract

We consider the one-dimensional heat and wave equations but -- instead of boundary conditions-- we 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.