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
DOI:
https://doi.org/10.1090/S0002-9947-98-02294-6
MathSciNet review:
1491875
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Abstract | References | Similar Articles | Additional Information
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 Poláčik 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
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