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September  2006, 6(5): 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

On a Burgers' type equation

Chun-Hsiung Hsia 1, and  Xiaoming Wang 2,

1. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, United States
2. 

Department of Mathematics, Florida State University, Tallahassee, FL32306, United States

Received  August 2005 Revised  January 2006 Published  June 2006

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.
Keywords: dimension of attractor., Burgers' type equation, stability, bifurcation, global attractor.
Mathematics Subject Classification: Primary: 37G35, 35B32; Secondary: 35B4.
Citation: Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121
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