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In this paper we study the dynamics of a Burgers' type equation (1).
First, we use a new method called attractor bifurcation introduced
by Ma and Wang in [4, 6] to study the bifurcation of Burgers' type
equation out of the trivial solution. For Dirichlet boundary
condition, we get pitchfork attrac- tor bifurcation as the parameter
$\lambda$
crosses the first eigenvalue. For periodic boundary condition, we
get bifurcated $S^{1}$
attractor consisting of steady states. Second, we
study the long time behavior of the equation. We show that there
exists a global attractor whose dimension is at least of the order
of $\sqrt{\lambda}$. Thus it provides another example of extended
system (see (2)) whose global attractor has a Hausdorff/fractal
dimension that scales at least linearly in the system size while the
long time dynamics is non-chaotic.