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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Complicated dynamics of parabolic equations
with simple gradient dependence


Authors: Martino Prizzi and Krzysztof P. Rybakowski
Journal: Trans. Amer. Math. Soc. 350 (1998), 3119-3130
MSC (1991): Primary 35K20; Secondary 35B40
MathSciNet review: 1491875
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Abstract: Let $\Omega \subset \mathbb R^{2}$ be a smooth bounded domain. Given positive integers $n$, $k$ and $q_{l}~\le ~l$, $l=1$, ..., $k$, consider the semilinear parabolic equation

\begin{alignat*}{2} u_{t}&=u_{xx}+u_{yy}+a(x,y)u+ \smash{\sum _{l=1}^{k}}a_{l}(x,y) u^{l-q_{l}}(u_{y})^{q_{l}},&\quad &t>0, (x,y)\in \Omega,\tag{E}\\ u&=0,&\quad& t>0, (x,y)\in \partial \Omega . \end{alignat*}

where $a(x,y)$ and $a_{l}(x,y)$ are smooth functions. By refining and extending previous results of Polácik we show that arbitrary $k$-jets of vector fields in $\mathbb R^{n}$ can be realized in equations of the form (E). In particular, taking $q_{l}\equiv 1$ we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on $\nabla u$.


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Additional Information

Martino Prizzi
Affiliation: SISSA, via Beirut 2-4, 34013 Trieste, Italy
Email: prizzi@sissa.it

Krzysztof P. Rybakowski
Affiliation: Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 18055 Rostock, Germany
Email: krzysztof.rybakowski@mathematik.uni-rostock.de

DOI: https://doi.org/10.1090/S0002-9947-98-02294-6
Keywords: Center manifolds, jet realization, parabolic equations, chaos.
Received by editor(s): May 16, 1996
Additional Notes: The research of the second author was supported, in part by MURST 40% and in part by the project Reaction-Diffusion Equations, Contract no. ERB CHRX CT 930 409, of the Human Capital and Mobility Programme of the European Communities
Article copyright: © Copyright 1998 American Mathematical Society