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Spectral theory and almost periodicity of mild solutions of non-autonomous Cauchy problems. (English) Zbl 0912.47022
Tübingen: Univ. Tübingen, Mathematische Fakultät, 82 p. (1997).
In the present thesis, we concentrate on linear non-autonomous Cauchy problems of the form $\dot u(t)= A(t)u(t)+f(t),\quad t\in J,\tag{1}$ where $$J$$ stands for $$\mathbb{R}$$ resp. $$\mathbb{R}^+$$, $$(A(t), D(A(t)))_{t\in J}$$ are linear operators on a Banach space $$X$$, and $$f$$ is a suitable function from $$J$$ into $$X$$.…
Assume that the homogeneous problem $$\dot u(t)= A(t)u(t)$$ is well-posed, i.e., the solutions yield an evolution family $${\mathcal U}= \{U(t, s): t\geq s\in J\}$$ of bounded linear operators on $$X$$. Then the evolution in time of system (1) is represented by $t\mapsto u(t)= U(t,s) u(s)+ \int^t_s U(t,\tau) f(\tau)d\tau,\quad t\geq s\in J.\tag{2}$ The purpose of this thesis is to deduce almost periodicity properties of the function $$u$$ from almost periodicity properties of the evolution family $${\mathcal U}$$ and the function $$f$$ in conjunction with spectral assumptions on operators connected with $${\mathcal U}$$. In other words, we discuss the following question.
Let $$U(t+\cdot, s+\cdot)x$$ be almost periodic for all $$t\geq s\in J$$ and $$x\in X$$. Suppose that $$f: J\to X$$ is almost periodic. What are suitable spectral conditions such that $$u$$ is almost periodic?…
We now sketch the results of this thesis. In Chapter 1, we discuss almost periodicity properties of bounded, uniformly continuous vector-valued functions $$t\mapsto f(t)$$ on the (half-)line. In particular, we focus on the behaviour of $$f$$ for large values of $$t$$ under assumptions on the behaviour of its Laplace, resp. Carleman transform.…
In Chapter 2, we turn to non-autonomous Cauchy problem. We are concerned with the asymptotic behaviour of the orbits of an evolution family satisfying certain almost periodic properties. For this purpose, we develop a spectral theory for evolution families and the induced evolution semigroups.…
Chapters 3 and 4 are devoted to inhomogeneous non-autonomous Cauchy problems. As above, we assume that the evolution family $${\mathcal U}$$ has certain almost periodicity properties and denote by $${\mathcal T}$$ the induced evolution semigroup, with generator $$G$$. In Chapter 3, we are concerned with the condition $$\sigma(G)\cap i\mathbb{R}= \emptyset$$.…
Finally, in Chapter 4, we are concerned with periodic evolution families $${\mathcal U}$$ employing the assumption $\sigma(G)\cap i\mathbb{R}\quad\text{ is countable}$ (and non-empty), where $$G$$ is the generator of the induced evolution semigroup on the space of periodic functions. As indicated above, such a condition cannot be expected for non-periodic evolution families. It turns out that $$\sigma(G)\cap i\mathbb{R}$$ is countable, if and only if $$\sigma(V)\cap \Gamma$$ is countable, where $$V$$ denotes the monodromy operator and $$\Gamma$$ is the unit circle in $$\mathbb{C}$$. If $$u$$ is given by (2), then under appropriate assumptions on $$u$$ we can show that almost periodicity and stability properties of the inhomogeneity $$f$$ are inherited by the mild solution $$u$$.
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces