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**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).

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 |