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Quasisolutions and global attractor of reaction-diffusion systems. (English) Zbl 0853.35056
The author considers a reaction-diffusion system ${\partial u_i\over \partial t}= L_i u_i+ f_i(x, u)\quad \text{in } \Omega\times (0, \infty),$ $B_i u_i= h_i\quad \text{on } \partial \Omega\times (0, \infty),\quad u_i(x, 0)= u_{i, 0}(x)\quad \text{in } \Omega,$ where $$u= (u_1,\dots, u_n)$$. The nonlinear term $$f$$ is called mixed quasimonotone if and only if for each $$i$$ there exist $$a_i$$ and $$b_i$$ such that $$a_i+ b_i= n- 1$$ and $$f$$ is monotone nondecreasing in $$a_i$$ components of $$(u_1, u_2,\dots, u_{i- 1}, u_{i+ 1},\dots, u_n)$$ and monotone nonincreasing in other $$b_i$$ components of that. For this special case the global attractor of the dynamics is characterized as a cone defined in the ordered function space.
Reviewer: S.Jimbo (Sapporo)

##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs
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