Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-98-02294-6
MathSciNet review: 1491875
Full-text PDF

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 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 $\nabla u$.


References [Enhancements On Off] (What's this?)

  • 1. E. N. Dancer, On the existence of two-dimensional invariant tori for scalar parabolic equations with time periodic coefficients, Annali Scuola Norm. Sup. Pisa 43 (1991), 455 - 471. MR 93c:35077
  • 2. E. N. Dancer and P. Polácik, Realization of vector fields and dynamics of spatially homogeneous parabolic equations, preprint.
  • 3. T. Faria and L. Magalhães, Realisation of ordinary differential equations by retarded functional differential equations in neighborhoods of equilibrium points, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 759-776. MR 97g:34088
  • 4. B. Fiedler and P. Polá cik, Complicated dynamics of scalar reaction-diffusion equations with a nonlocal term, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 167-192. MR 91g:45008
  • 5. B. Fiedler and B. Sandstede, Dynamics of periodically forced parabolic equations on the circle, Ergodic Theory Dynamical Systems 12 (1992), 559-571. MR 93h:35103
  • 6. J. K. Hale, Flows on centre manifolds for scalar functional differential equations, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 193-201. MR 87d:34117
  • 7. P. Polácik, Complicated dynamics in scalar semilinear parabolic equations in higher space dimension, J. Differential Equations 89 (1991), 244-271. MR 92c:35063
  • 8. P. Polácik, Imbedding of any vector field in a scalar semilinear parabolic equation, Proc. Amer. Math. Soc. 115 (1992), 1001-1008. MR 92j:35099
  • 9. P. Polácik, Realization of any finite jet in a scalar semilinear equation on the ball in $\mathbb R^{3}$, Ann. Scuola Norm. Sup. Pisa XVII (1991), 83-102. MR 92g:35107
  • 10. P. Polácik, Realization of the dynamics of ODEs in scalar parabolic PDEs, Tatra Mountains Math. Publ. 4 (1994), 179-185. MR 95f:35117
  • 11. P. Polácik, High-dimensional $\omega $-limit sets and chaos in scalar parabolic equations, J. Differential Equations 119 (1995), 24-53. MR 96h:35092
  • 12. P. Polácik, Reaction-diffusion equations and realization of gradient vector fields, Proc. Equa- diff '95 (to appear).
  • 13. P. Polácik and K. P. Rybakowski, Imbedding vector fields in scalar parabolic Dirichlet BVPs, Ann. Scuola Norm. Sup. Pisa XXI (1995), 737-749. MR 97a:35124
  • 14. P. Polácik and K. P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations 124 (1995), 472- 494. MR 96m:35176
  • 15. M. Prizzi and K. P. Rybakowski, Inverse problems and chaotic dynamics of parabolic equations on arbitrary spatial domains, J. Differential Equations 142 (1998), 17-53.
  • 16. K. P. Rybakowski, Realization of arbitrary vector fields on center manifolds of parabolic Dirichlet BVPs, J. Differential Equations 114 (1994), 199-221. MR 95j:35114
  • 17. K. P. Rybakowski, Realization of arbitrary vector fields on invariant manifolds of delay equations, J. Differential Equations 114 (1994), 222-231. MR 96e:34121
  • 18. K. P. Rybakowski, The center manifold technique and complex dynamics of parabolic equations, Topological Methods in Differential Equations and Inclusions (A. Granas, M. Frigon, eds.), NATO ASI Series, vol. 472, Kluwer Academic Publishers, Dordrecht/Boston/London, 1995, pp. 411-446. MR 97b:35095

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K20, 35B40

Retrieve articles in all journals with MSC (1991): 35K20, 35B40


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

American Mathematical Society