Asymptotical dynamics of the modified Schnackenberg equations

Yuncheng You

doi:10.3934/proc.2009.2009.857

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2009, Volume 2009: 857-868. Doi: 10.3934/proc.2009.2009.857 open access

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Asymptotical dynamics of the modified Schnackenberg equations

Yuncheng You^1,

1.
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620

Received: May 31, 2008

Revised: January 31, 2009

Published: August 2009

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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.

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Mathematics Subject Classification: 37L30, 35B40, 35B41, 35K55, 35K57, 35Q80.

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