Dynamics of Partial Differential Equations

Volume 3 (2006)

Number 3

Large diffusivity stability of attractors in the C(Ω) — topology for a semilinear reaction and diffusion system of equations

Pages: 173 – 197

DOI: http://dx.doi.org/10.4310/DPDE.2006.v3.n3.a1

Author

Robert Willie (Department of Mathematics, University of Zimbabwe, Harare, Zimbabwe)

Abstract

A semilinear system of reaction and diffusion equations with nonlinear boundary conditions and discontinuous real valued data for which a mild solution is only known to exist is studied from the point of view of the Hölder continuity of the solutions. This regularity of the solutions furnishes the stability of attractors in an adequate notion via Arzela-Ascoli’s Theorem uniformly on the domain in arbitrary space dimensions. The limit in question is of large diffusivity. In this case, the dynamics of the limit process are governed by a nonlinear coupled system of ordinary differential equations. This allows to assert that in chemical engineering or biochemical reactions models, the long time dynamics of systems of reaction and diffusion equations yielding the formation of spatially heterogeneous stable states of patterns of concentrations by certain chemical substances can only occur in the case of relatively small diffusions.

Keywords

reaction and diffusion equations, global attractor, nonlinear boundary conditions

2010 Mathematics Subject Classification

35B25, 35B40, 35B45, 35Bxx

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