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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Complicated dynamics of parabolic equations with simple gradient dependence
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by Martino Prizzi and Krzysztof P. Rybakowski PDF
Trans. Amer. Math. Soc. 350 (1998), 3119-3130 Request permission


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:
  • Krzysztof P. Rybakowski
  • Affiliation: Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 18055 Rostock, Germany
  • Email:
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 3119-3130
  • MSC (1991): Primary 35K20; Secondary 35B40
  • DOI:
  • MathSciNet review: 1491875