On classical solvability for a linear 1D heat equation with constant delay

Abstract

In this paper, we consider a linear heat equation with constant coefficients and a single constant delay. Such equations are commonly used to model and study various problems arising in ecology and population biology when describing the temporal evolution of human or animal populations accounting for migration, interaction with the environment and certain aftereffects caused by diseases or enviromental polution, etc. Whereas dynamical systems with lumped parameters have been addressed in numerous investigations, there are still a lot of open questions for the case of systems with distributed parameters, especially when the delay effects are incorporated. In the present paper, we consider a general non-homogeneous one-dimensional heat equation with delay in both higher and lower order terms subject to non-homogeneous initial and boundary conditions. For this, we prove a unique existence of a classical solution as well as its continuous dependence on the data.