## Strongly order-preserving local semi-dynamical systems - theory and applications.(English)Zbl 0634.34031

Semigroups, theory and applications, Vol. 1, Pap. Autumn Course Int. Cent. Theor. Phys., Miramare (Trieste)/Italy 1984, Pitman Res. Notes Math. Ser. 141, 178-189 (1986).
[For the entire collection see Zbl 0585.00006.]
The paper contains the very interesting notion of a strongly order- preserving local semiflow, which allows one to generalize some results concerning the asymptotic behaviour of the solutions of nonlinear evolution equations due to M. Hirsch [Bull. Am. Math. Soc., New Ser. 11, 1-6 (1984; Zbl 0541.34026)] as well as such ones of the author [J. Fac. Sci., Univ. Tokyo Sect. IA 30, 645-673 (1984; Zbl 0545.35042)]. Let $$\Phi =(\Phi_ t)_{t\geq 0}$$ be a local semiflow on the complete metric ordered space X. $$\Phi$$ is called strongly order-preserving if for any two points x,y$$\in X$$ with $$x>y$$ and any $$t\in]0$$, min(t $$+(x),t$$ $$+(y))[$$ there exist open neighbourhoods V of x, W of y such that $$\phi_ t(V)>\Phi_ t(W)$$. This notion is more general than the corresponding one of Hirsch, since the latter requires that the order structure is defined by a closed cone having nonempty interior.
Among the vaious interesting results we mention the following one: Let $$\Phi$$ be strongly order-preseving and let $$\Phi$$ satisfy some compactness condition. Suppose Y is a bounded, positively invariant, closed, stable subset. Then Y contains at least one stable equilibrium point. Among the applications we mention the following system: $\dot u=d_ 1\Delta u+f(u,c),\quad \dot v=d_ 2\Delta u+g(u,v),$ where $$d_ 1\neq d_ 2$$ and $$f_ v<0$$, $$g_ u<0$$ on the region $$\Omega$$ under consideration. The paper contains no proofs. These will appear in a not yet published book (“Asymptotic behavior of nonlinear diffusion equations ”, Research Notes in Math. Pitman).
Reviewer: M.Wolff

### MSC:

 37-XX Dynamical systems and ergodic theory 34G20 Nonlinear differential equations in abstract spaces 35K55 Nonlinear parabolic equations 35K65 Degenerate parabolic equations

### Citations:

Zbl 0585.00006; Zbl 0541.34026; Zbl 0545.35042