Complicated dynamics of parabolic equations with simple gradient dependence

Let Omega subset of R-2 be a smooth bounded domain. Given positive integers n, k and ql less than or equal to l, l = 1,... k, consider the semilinear parabolic equation [GRAPHICS] where a(x, y) and a(l)(x, y) are smooth functions. By refining and extending previous results of Polacik we show that arbitrary k-jets of vector fields in R-n can be realized in equations of the form (E). In particular, taking q(l) = 1 we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on del u.