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Reaction-diffusion with memory in the minimal state framework


Authors: Monica Conti, Elsa M. Marchini and Vittorino Pata
Journal: Trans. Amer. Math. Soc. 366 (2014), 4969-4986
MSC (2010): Primary 35B41, 35K05, 45K05, 47H20
DOI: https://doi.org/10.1090/S0002-9947-2013-06097-7
Published electronically: November 6, 2013
MathSciNet review: 3217706
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Abstract: We consider the integrodifferential equation

$\displaystyle \partial _t u-\Delta u -\int _0^\infty \kappa (s)\Delta u(t-s)\,\textup {d} s + \varphi (u)=f$

arising in the Coleman-Gurtin theory of heat conduction with hereditary memory. Within a novel abstract framework, based on the notion of minimal state, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semigroup of solutions.

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Additional Information

Monica Conti
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
Email: monica.conti@polimi.it

Elsa M. Marchini
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
Email: elsa.marchini@polimi.it

Vittorino Pata
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
Email: vittorino.pata@polimi.it

DOI: https://doi.org/10.1090/S0002-9947-2013-06097-7
Keywords: Heat conduction with memory, minimal state framework, solution semigroup, global attractor, exponential attractor
Received by editor(s): October 17, 2011
Received by editor(s) in revised form: February 1, 2013
Published electronically: November 6, 2013
Article copyright: © Copyright 2013 American Mathematical Society