Periodic processes and non-autonomous evolution equations with time- periodic terms
The starting point in the authors' consideration is a Cauchy type problem \[ u_ t= A(u, t), \tag{1} \] where \(A\) is a time dependent mapping of some Banach space \(E\) into itself, subject to some constraints; among others it is \(p\)-periodic i.e. \(A(u, t+ p)= A(u, t)\). It is assumed that the solution to (1) is described by a family \(U(t,\tau)\), \(t\geq \tau\) of (nonlinear) mappings from \(E\) into itself, such that \(u(t)= U(t, \tau) u(\tau)\) solves the initial value problem (1) with \(u(\tau)\) the initial value at \(t= \tau\); under the periodicity condition also(2) \(U(t, \tau)= U(t+ p, \tau+ p)\) holds. In order to study attractors and related concepts of the flow generated by (1) the authors adopt a path which differs from the familiar approach via Poincaré map. They define a semigroup \(S(t)\), \(t\in \mathbb{R}\) on \(E\times T^ 1\), with \(T^ 1\) the circle with length \(p\), according to \[ S(t)(u,\delta)= (U(\delta+ t) u, \delta+ t\text{ mod } p).\tag{2} \] Attractors are then studied in terms of \(S(t)\). In an introductory section, the authors disregard the Cauchy problem (1) and study suitable families \(U(t, \tau)\), \(t\geq \tau\) in their own right. They call a family of mappings \(U(t, \tau)\), \(t\geq \tau\) from \(E\) into \(E\) a ``process'' if, besides some continuity assumptions, the usual causality assumption is satisfied; if moreover (2) holds, the process is called \(p\)-periodic. More generally it is admitted that the process depends on a parameter \(\sigma\) which varies over a metrical space \(\Sigma\), subject to some conditions. In particular a semigroup \(T(t)\), \(t\geq 0\) is given which acts on \(\Sigma\). It is assumed that the following relation holds: \[ U_ \sigma(t+ s, \tau+ s)= U_{T(s)\sigma}(t, \tau).\tag{3} \] Under this assumption a semigroup \(S(t)\), operating on \(E\times \Sigma\), is defined according to \[ S(t)(u, \sigma)= (U_ \sigma(t, 0) u, T(t)\sigma).\tag{4} \] Under certain continuity and compactness assumptions the authors can prove a number of theorems concerning the existence of attractors related to the semigroup \(S(t)\). In a further section, fractal and Hausdorff dimension of these attractors are estimated, and in a final section, applications to concrete problems are given.
scientific article; zbMATH DE number 733356