Asymptotical dynamics of the modified Schnackenberg equations. (English) Zbl 1191.37043

You, Yuncheng

Discrete Contin. Dyn. Syst. 2009, Suppl., 857-868 (2009).

Let \(\Omega \subset \mathbb{R}^n\; (n\leq 3)\) be a bounded domain that has a locally Lipschitz continuous boundary and lies locally on one side of its boundary. The author studies the asymptotical dynamics of the modified Schnackenberg equations (MSE) on \(\Omega,\) \[ \frac{\partial u}{\partial t}=d_1 \Delta u+\rho-au+u^2v-Fu^3,\quad t>0,\;x\in \Omega \eqno({1}) \] \[ \frac{\partial v}{\partial t}=d_2 \Delta v+b -u^2v+Fu^3,\quad t>0,\;x\in \Omega \eqno({2}) \] where the parameters \(d_1,d_2,\rho, a,b\) and \(F\) are positive constants, with the homogeneous Dirichlet (non-slip) boundary conditions \[ u(x,t)=0\;\;v(x,t)=0\;\;t>0,\;x\in \partial \Omega \eqno({3}) \] and initial conditions \[ u(0,t)=u_0(x)\;\;v(0,t)=v_0(x)\;\;\;x\in \Omega \eqno({4}) \] without assumption that initial data \(u_0\) and \(v_0\), nor \(u(x,t)\) and \(v(x,t)\), are nonnegative.
The existence of a global attractor in the \(L^2\) product phase space for the solution semiflow for this simplified model of some trimolecular autocatalytic biochemical or chemical reactions with diffusion is proved. Applications of this model of equations including pattern formations in embryogenesis and skin analysis are known.

Reviewer: Irina V. Konopleva (Ul’yanovsk)

MSC:

37L30	Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35B40	Asymptotic behavior of solutions to PDEs
35B41	Attractors
35K55	Nonlinear parabolic equations
35K57	Reaction-diffusion equations
35Q92	PDEs in connection with biology, chemistry and other natural sciences

Keywords:

Schnackenberg equation; global dynamics; global attractor; absorbing set; asymptotical compactness