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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 10, Issue 3, 1991, pp. 275–317**

**DOI: 10.4171/ZAA/452**

Published online: 1991-09-30

Domains of Attraction of Generic $\omega$-Limit Sets for Strongly Monotone Semiflows

Peter TakácAsymptotic behavior of a strongly increasing semiflow $\Omega$ in a strongly ordered metrizable topological space $X$ is investigated in terms of the $\omega$-limit set $\omega (x)$ of a generic point $x \in X$ whose positive semiorbit $\mathcal O^+ (x)$ is assumed to be relatively compact. The domain of attraction of the $\omega$-limit set of a generic order $\omega$-stable point is determined. If $X$ is an open and order-convex subset of a separable strongly ordered Banach space $V$, it is proved that "almost all" points $x \in X$ are order $\omega$-stable, whereas the remaining $\omega$-unstable points are contained in the union of at most countably many Lipschitz manifolds of codimension one in $V$. If $\Omega$ admits a strongly positive, compact linearization about its equilibria, then $\omega (x)$ is a single equilibrium for every order $\omega$-stable point $x \in X$.

*Keywords: *convergence to equilibrium, invariant order decomposition and resolution, lower and upper $\omega$-limit sets, cooperative system of ordinary differential equations, parabolic partial differential equation

Takác Peter: Domains of Attraction of Generic $\omega$-Limit Sets for Strongly Monotone Semiflows. *Z. Anal. Anwend.* 10 (1991), 275-317. doi: 10.4171/ZAA/452