Dynamics of Partial Differential Equations

Volume 10 (2013)

Number 3

Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay

Pages: 267 – 281

DOI: https://dx.doi.org/10.4310/DPDE.2013.v10.n3.a3

Authors

T. Caraballo (Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain)

A. M. Márquez–Durán (Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain; Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, Sevilla, Spain)

Abstract

A nonclassical nonautonomous diffusion equation with delay is analyzed. First, we prove the existence and uniqueness of solutions by using the Galerkin approximations and the energy method. Next, we prove the existence and eventual uniqueness of stationary solutions, as well as their exponential stability. We emphasize that the assumptions imposed on the delay term include, in particular, the case of measurable variable delays.

Keywords

diffusion equation with delay, existence and uniqueness

2010 Mathematics Subject Classification

35-xx

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Published 11 October 2013