On a Burgers' type equation

Pages: 1121 - 1139, Volume 6, Issue 5, September 2006

doi:10.3934/dcdsb.2006.6.1121       Abstract        Full Text (293.4K)       Related Articles

Chun-Hsiung Hsia - Department of Mathematics, Indiana University, Bloomington, IN 47405, United States (email)
Xiaoming Wang - Department of Mathematics, Florida State University, Tallahassee, FL32306, United States (email)

Abstract: 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:  Burgers' type equation, bifurcation, stability, global attractor, dimension of attractor.
Mathematics Subject Classification:  Primary: 37G35, 35B32; Secondary: 35B40.

Received: August 2005;      Revised: January 2006;      Published: June 2006.