On a Burgers’ Type Equation

In this paper we study the dynamics of a Burgers’ type equation (1.1). 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 attractor bifurcation as the parameter λ crosses the first eigenvalue. For periodic boundary condition, we get bifurcated S 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 √ λ. Thus it provides another example of extended system (see (1.2)) 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.