Complicated dynamics of parabolic equations with simple gradient dependence期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Complicated dynamics of parabolic equations with simple gradient dependence

Authors:	Martino Prizzi Krzysztof P. Rybakowski

Affiliation:	SISSA, via Beirut 2-4, 34013 Trieste, Italy ; Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 18055 Rostock, Germany

Abstract:	Let $Omega subset mathbb R^{2}$ be a smooth bounded domain. Given positive integers , and $q_{l}~le ~l$ , , ..., , 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 and $a_{l}(x,y)$ are smooth functions. By refining and extending previous results of Polácik we show that arbitrary -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 .

Keywords:	Center manifolds jet realization parabolic equations chaos.

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