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
Full-text PDF Free Access
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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 $\varepsilon \to 0$. 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.
References
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References
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- A. Babin, M.I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam, 1992. MR 1156492 (93d:58090)
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- J. Fort, V. Méndez, Wavefront in time-delayed reaction-diffusion systems. Theory and comparison to experiments, Rep. Prog. Phys. 65, 895–954, 2002.
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- S. Gatti, V. Pata, A one-dimensional wave equation with nonlinear damping, Glasg. Math. J. 48, 419–430, 2006. MR 2271373 (2007k:35331)
- G. Gentili and C. Giorgi, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math. 51, 342–362, 1993. MR 1218373 (94j:80004)
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- C. Giorgi, V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, NoDEA Nonlinear Differential Equations Appl. 8, 157–171, 2001. MR 1834457 (2002d:35210)
- M. Grasselli, V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory, Indiana University Mathematics Journal 50, 1281–1308, 2001. MR 1871356 (2002h:35031)
- M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp. 155–178, Progr. Nonlinear Differential Equations, Appl. no.50, Birkhäuser, Boston, 2002. MR 1944162 (2003j:37135)
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- M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113–126, 1968. MR 1553521
- J.K. Hale, Asymptotic behaviour of dissipative systems, Amer. Math. Soc., Providence, RI, 1988. MR 941371 (89g:58059)
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- T. Hillen, Qualitative analysis of semilinear Cattaneo equations, Math. Models Methods Appl. Sci. 8, 507–519, 1998. MR 1624816 (99c:35121)
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- V. Méndez, J. Fort, J. Farjas, Speed of wavefront solutions to hyperbolic reaction-diffusion systems, Phys. Rev. E 60, 5231–5243, 1999. MR 1719397 (2000h:35094)
- V. Méndez, J.E. Llebot, Hyperbolic reaction-diffusion equations for a forest fire model, Phys. Rev. E 56, 6557–6563, 1997. MR 1492398
- J.W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29, 187–204, 1971. MR 0295683 (45:4749)
- W.E. Olmstead, S.H. Davis, S. Rosenblat, W.L. Kath, Bifurcation with memory, SIAM J. Appl. Math. 46, 171–188, 1986. MR 833472 (87f:35020)
- V. Pata, S. Zelik, Attractors and their regularity for 2-D wave equation with nonlinear damping, Adv. Math. Sci. Appl. 17, 225–237, 2007. MR 2337377 (2008j:37168)
- V. Pata, S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping, Math. Methods Appl. Sci. 29, 1291–1306, 2006. MR 2247700 (2007h:37127)
- V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11, 505–529, 2001. MR 1907454 (2003f:35027)
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
- R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York, 1988. MR 953967 (89m:58056)
- S. Zheng, Nonlinear evolution equations, Chapman & Hall/CRC, Boca Raton, 2004. MR 2088362 (2006a:35001)
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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
MR Author ID:
358540
Email:
vittorino.pata@polimi.it
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.