Asymptotical dynamics of the modified Schnackenberg equations

Pages: 857 - 868, Issue Special, September 2009

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Yuncheng You - Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, United States (email)

Abstract: The existence of a global attractor in the $L^2$ product phase space for the solution semiflow of the modified Schnackenberg equations with the Dirichlet boundary condition on a bounded domain of space dimension $n\le 3$ is proved. This reaction-diffusion system features two pairs of oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The proof features two types of rescaling and grouping estimation in showing the absorbing property and the uniform smallness in proving the asymptotical compactness by the approach of a new decomposition.

Keywords:  Schnackenberg equation, global dynamics, global attractor, absorbing set, asymptotical compactness
Mathematics Subject Classification:  37L30, 35B40, 35B41, 35K55, 35K57, 35Q80

Received: June 2008;      Revised: February 2009;      Published: September 2009.