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On monotone trajectories. (English) Zbl 0737.58010

For a metric space \(X\) a semiflow on \(X\) is called a family of continuous mappings \(\Phi_ t: [0,\infty)\times X\to X\) satisfying the following: 1) \(\Phi_ 0=I_ X\), 2) \(\Phi_ t(\Phi_ s(x))=\Phi_{t+s}(x)\) for \(t,s\in [0,\infty)\). A semiflow on an open subset of a strongly orderd Banach space \(V\) is said to be strongly monotone if, for each \(t>0\), \(x,y\in X\), the inequality \(x<y\) implies \(\Phi_ t(x)\ll\Phi_ t(y)\). Let \(0\in X\) be an equilibrium for a strongly monotone semiflow \(\Phi_ t\). \(K(\varepsilon)\) denotes the closed ball of center 0 and radius \(\varepsilon>0\), \(B(\varepsilon):=\{x\in K(\varepsilon): |\Phi_ t(x)|\to 0\hbox{ as }t\to\infty\}\), \(B_ +(\varepsilon)\) (resp. \(B_ -(\varepsilon)\)) \(:=\{x\in B(\varepsilon)\); the trajectory of \(x\) is eventually strongly decreasing (resp. increasing)}, \(B_ 0(\varepsilon):=B(\varepsilon)\backslash(B_ +(\varepsilon)\cup B_ - (\varepsilon))\), \(W_ 2(\varepsilon)\) is the local strongly stable invariant \(C^ 1\) manifold tangent at 0 to the subspace \(V_ 2\) in the Krein-Rutman decomposition for \(D\Phi_ 1(0)\). Standing hypothesis: 0 is an equilibrium for a \(C^ 1\) strongly monotone semiflow \(\Phi_ t\) such that \(D\Phi_ 1(0)\) is compact with spectral radius \(\varrho\leq 1\).
Theorem 1. There exists \(\varepsilon>0\) such that \(B_ 0(\varepsilon)\subset W_ 2(\varepsilon)\) and \(B_ 0(\varepsilon)\) is nowhere dense in \(K(\varepsilon)\).
Let be \(\Sigma_ +(\varepsilon):=\hbox{(resp. }\Sigma_ - (\varepsilon):=)\{x\in B(\varepsilon): \exists T>0\hbox{ such that }\Phi_ t(x)\gg 0\hbox{ (resp. }\Phi_ t(x)\ll 0)\hbox{ for all }t>T\}\), and \(\Sigma_ 0(\varepsilon):=B(\varepsilon)\backslash(\Sigma_ +(\varepsilon)\cup\Sigma_ -(\varepsilon))\).
Theorem \(1'\). There exists an \(\varepsilon>0\) such that \(\Sigma_ 0(\varepsilon)\subset W_ 2(\varepsilon)\).
Theorem 2. If the set \(X_ 0\) of points with precompact orbits is dense in \(K(\varepsilon)\) and for each \(x\in B(\varepsilon)\backslash\{0\}\) \(\inf\{\mu>0: \lim\inf_{t\to\infty}| \Phi_ t(x)|/\mu^ t=0\}>0\), then \(\Sigma_ 0(\varepsilon)=W_ 2(\varepsilon)\).

MSC:

58D07 Groups and semigroups of nonlinear operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H20 Semigroups of nonlinear operators
37C75 Stability theory for smooth dynamical systems
35B40 Asymptotic behavior of solutions to PDEs
35K10 Second-order parabolic equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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