COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS GLOBAL ATTRACTOR OF THE GRAY-SCOTT EQUATIONS Yuncheng You Department of Mathematics and Statistics University
Trang 1COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS
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 ≤ 3 is proved This
reaction-diffusion system does not have dissipative property inherently due to the
op-positely 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.
1 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
∂t = d1∆u − (F + k)u + u
2v, t > 0, x ∈ Ω, (1.1)
∂v
∂t = d2∆v + F (1 − v) − u
2v, t > 0, x ∈ Ω, (1.2) with the homogeneous Neumann (non-flux) boundary conditions
∂u
∂ν(t, x) = 0,
∂v
∂ν(t, x) = 0, t > 0, x ∈ ∂Ω, (1.3) 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 do 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 autocat-alytic, continuously fed, unstirred reaction and diffusion of two chemicals with con-centrations 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.
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