Well-Posedness and Asymptotic Behavior of a Nonclassical Nonautonomous Diffusion Equation with Delay

Author:

Caraballo Tomás1,Márquez-Durán Antonio M.2,Rivero Felipe3

Affiliation:

1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain

2. Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1, 41013-Sevilla, Spain

3. Departamento de Análise, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga S/N Valonguinho, 24.020-140 Niteró, RJ, Brazil

Abstract

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzelà theorem to prove the compactness.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modeling and Simulation,Engineering (miscellaneous)

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