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On a Burgers' type equation

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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.
    Mathematics Subject Classification: Primary: 37G35, 35B32; Secondary: 35B40.


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