Global attractor of the gray scott equations

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 7, Number 4, July 2008 Website: http://AIMsciences.org pp 947–970 GLOBAL ATTRACTOR OF THE GRAY-SCOTT EQUATIONS Yuncheng You Department of Mathematics and Statistics University of South Florida, Tampa, FL 33620, USA Abstract In this work the existence of a global attractor for the solution semiflow of the Gray-Scott equations with the Neumann boundary conditions on bounded domains of space dimensions n ≤ is proved This reactiondiffusion system does not have dissipative property inherently due to the oppositely signed nonlinearity The asymptotical compactness is shown by a new decomposition method It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite Introduction Chemical and biochemical kinetics has been a rich source to produce a variety of spatial-temporal patterns since the discovery of the oscillating wave in the Belousov-Zhabotinsky reaction [1, 31] in 1950’s These phenomena and observations have been transferred to challenging mathematical problems through various mathematical models, especially reaction-diffusion equations Among these mathematical models, typical are the autocatalytic models of glycolysis by Sel’kov [26, 25] and for isothermal systems by Gray-Scott [4, 5, 6, 2] In this paper, we shall study the global dynamics of the following Gray-Scott equations on a bounded domain Ω ⊂ ℜn (n ≤ 3) which has a locally Lipschitz continuous boundary and lies locally on one side of its boundary, ∂u = d1 ∆u − (F + k)u + u2 v, t > 0, x ∈ Ω, (1.1) ∂t ∂v = d2 ∆v + F (1 − v) − u2 v, t > 0, x ∈ Ω, (1.2) ∂t with the homogeneous Neumann (non-flux) boundary conditions ∂u ∂v (t, x) = 0, (t, x) = 0, t > 0, x ∈ ∂Ω, (1.3) ∂ν ∂ν where d1 , d2 , F , and k are positive constants, and ∂/∂ν is the outward normal derivative, and with a given initial condition u(0, x) = u0 (x), v(0, x) = v0 (x), x ∈ Ω (1.4) We not assume initial data u0 and v0 are nonnegative and/or bounded Thus the solutions (u, v) are not necessarily nonnegative The Gray-Scott system was originated from modeling an isothermal autocatalytic, continuously fed, unstirred reaction and diffusion of two chemicals with concentrations u(t, x) and v(t, x), see [4, 5, 6, 7, 23] The well-known examples of 2000 Mathematics Subject Classification 37L30, 35B40, 35B41, 35K55, 35K57, 35Q80 Key words and phrases Gray-Scott equation, global attractor, global dynamics, absorbing set, asymptotic compactness, fractal dimension 947