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Mat. Sb., 2009, Volume 200, Number 4, Pages 3–30 (Mi msb7298)  

This article is cited in 6 scientific papers (total in 6 papers)

Trajectory attractors of reaction-diffusion systems with small diffusion

M. I. Vishik, V. V. Chepyzhov

A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: We consider a reaction-diffusion system of two equations, where one equation has a small diffusion coefficient $\delta>0$. We construct the trajectory attractor $\mathfrak A^\delta$ of such a system. We also study the limit system for $\delta=0$. In this system one equation is an ordinary differential equation in $t$, but is considered in the domain $\Omega\times\mathbb R_+$, where $\Omega\Subset\mathbb R^n$ and $\mathbb R_+$ is the positive time axis, $t$. We construct the trajectory attractor $\mathfrak A^0$ of the limit system. The main result is a convergence theorem: $\mathfrak A^\delta\to\mathfrak A^0$ as $\delta\to0^+$ in the corresponding topology.
Bibliography: 18 titles.

Keywords: trajectory attractor, reaction-diffusion equations.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm7298

Full text: PDF file (668 kB)
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English version:
Sbornik: Mathematics, 2009, 200:4, 471–497

Bibliographic databases:

UDC: 517.956.4
MSC: 35K57, 35B41
Received: 09.09.2008

Citation: M. I. Vishik, V. V. Chepyzhov, “Trajectory attractors of reaction-diffusion systems with small diffusion”, Mat. Sb., 200:4 (2009), 3–30; Sb. Math., 200:4 (2009), 471–497

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Chepyzhov V.V., Vishik M.I., “Trajectory attractor of a reaction-diffusion system with a series of zero diffusion coefficients”, Russ. J. Math. Phys., 16:2 (2009), 208–227  crossref  mathscinet  zmath  isi  elib  scopus
    2. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractor for a system of two reaction-diffusion equations with diffusion coefficient $\delta(t)\to 0+$ as $t\to+\infty$”, Dokl. Math., 81:2 (2010), 196–200  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. Chepyzhov V.V., Vishik M.I., “Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time”, Discrete Contin. Dyn. Syst., 27:4 (2010), 1493–1509  crossref  mathscinet  zmath  isi  elib  scopus
    4. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractors of equations of mathematical physics”, Russian Math. Surveys, 66:4 (2011), 637–731  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. M. Kh. Numan Elsheikh, D. O. Ogun, Yu. N. Orlov, R. V. Pleshakov, V. Zh. Sakbaev, “Usrednenie sluchainykh polugrupp i neodnoznachnost kvantovaniya gamiltonovykh sistem”, Preprinty IPM im. M. V. Keldysha, 2014, 019, 28 pp.  mathnet
    6. L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, Theoret. and Math. Phys., 185:2 (2015), 1582–1598  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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