Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Two-dimensional reaction-diffusion equations with memory

Authors: Monica Conti, Stefania Gatti, Maurizio Grasselli and Vittorino Pata
Journal: Quart. Appl. Math. 68 (2010), 607-643
MSC (2000): Primary 45K05; Secondary 35B40, 35B41, 76A10, 92D25
DOI: https://doi.org/10.1090/S0033-569X-2010-01167-7
Published electronically: September 17, 2010
MathSciNet review: 2761507
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Abstract: In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonincreasing summable memory kernel $ k$. This equation models several phenomena arising from many different areas. After rescaling $ k$ by a relaxation time $ \varepsilon>0$, we formulate a Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping, for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system which is dissipative provided that $ \varepsilon$ is small enough, namely, when the equation is sufficiently ``close'' to the standard reaction-diffusion equation formally obtained by replacing $ k$ with the Dirac mass at 0. Then, we provide an estimate of the difference between $ \varepsilon$-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit $ \varepsilon\to0$. In particular, this yields the existence of a regular global attractor of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework.

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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

Stefania Gatti
Affiliation: Dipartimento di Matematica, Università di Modena e Reggio Emilia via Campi 213/B, 41100 Modena, Italy
Email: stefania.gatti@unimore.it

Maurizio Grasselli
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano Via Bonardi 9, 20133 Milano, Italy
Email: maurizio.grasselli@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/S0033-569X-2010-01167-7
Keywords: Reaction-diffusion equations, memory effects, nonlinear damping, exponential attractors, global attractors, Lyapunov functionals, convergence to equilibria
Received by editor(s): January 13, 2009
Published electronically: September 17, 2010
Additional Notes: This work was partially supported by the Italian PRIN Research Project 2006 Problemi a frontiera libera, transizioni di fase e modelli di isteresi
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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