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

##### 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
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