Complicated dynamics of parabolic equations with simple gradient dependence

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 $\nabla u$.