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 \(u\).

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 |