Robust exponential attractors for population dynamics models withinfinite time delay

Cecilia Cavaterra; M. Grasselli

doi:10.3934/dcdsb.2006.6.1051

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2006, Volume 6, Issue 5: 1051-1076. Doi: 10.3934/dcdsb.2006.6.1051

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Robust exponential attractors for population dynamics models with infinite time delay

Cecilia Cavaterra^1, and
M. Grasselli^2,

1.
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via C. Saldini, 50, I-20133 Milano
2.
Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano

Received: October 31, 2005

Revised: December 31, 2005

Published: August 2006

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Abstract

We consider an integrodifferential reaction-diffusion system which finds application in population dynamics. The memory kernels accounting for delay effects can be of both weak and strong type. Rescaling the kernels with a time relaxation $\varepsilon>0$, we show that the original model gives raise to a one-parameter family of dynamical systems in a suitable phase-space, We prove that this family is characterized by a corresponding family of exponential attractors which is stable as the delay effects vanish, i.e., when $\varepsilon$ goes to $0$.

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Mathematics Subject Classification: Primary: 35B41, 92D25; Secondary: 37L25, 45K05.

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