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Global dynamics of the Brusselator equations. (English) Zbl 1158.37028
The author considers the following system of two nonlinearly coupled reaction-diffusion equations $${\partial u\over\partial t}= d_1\Delta_x u+ u^2v- (b+ 1)u+ a,\quad (t,x)\in (0,\infty)\times\Omega,\tag1$$ $${\partial v\over\partial t}= d_2\Delta_x v- u^2 v+ bu,\quad (t,x)\in (0,\infty)\times\Omega,\tag2$$ $$u(t,x)= v(t,x)= 0,\quad t> 0,\ x\in\partial\Omega,\tag3$$ $$u(0,x)= u_0(x),\ v(0,x)= v_0(x),\quad x\in\Omega,\tag4$$ where $\Omega$ is a bounded domain of $\Bbb R^d$, $d\le 3$, with locally Lipschitz continuous boundary, $d_1$, $d_2$, $a$ and $b$ are positive constants. For the initial boundary value problems (1)--(4), the essential difficulties in proving existence of a global attractor lie in the fact that the oppositely interactive polynomial nonlinearity in the two coupled equations does not possess partial dissipativity or asymptotic dissipativity. In this paper, a new decomposition technique is explored and used to show the $K$-contraction of the solution semiflow.

37L30Attractors and their dimensions, Lyapunov exponents
35B41Attractors (PDE)
35K57Reaction-diffusion equations
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