SUFFICIENT CONDITIONS FOR EXPONENTIAL STABILITY AND DICHOTOMY OF EVOLUTION EQUATIONS

ROLAND SCHNAUBELT

Abstract. We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u′(t) = A(t)u(t) (∗) on a Banach space X. Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A(·) has a sufficiently small Hölder constant, then (∗) has exponential dichotomy. We further study robustness of exponential dichotomy under time dependent unbounded Miyadera-type perturbations. Our main tool is a characterization of exponential dichotomy of evolution families by means of the spectra of the so-called evolution semigroup on C_{0}(ℝ, X) or L^{1}(ℝ, X).

1. Introduction and preliminaries

Exponential dichotomy is one of the fundamental asymptotic properties of solutions of the linear Cauchy problem

(CP) \{ \begin{matrix}\frac{d}{dt}u(t) & = & A(t)u(t), t > s, \\ u(s) & = & x \end{matrix}

in a Banach space X. It also plays an important role in the investigation of qualitative properties of nonlinear evolution equations such as linearized (in-)stability or the existence of center manifolds, see e.g. [8, 12, 19, 32] and the references therein.

In the autonomous case, i.e., A(t) ≡ A, exponential dichotomy is charcaterized by the spectral condition σ(A) ∩ iℝ = ∅ provided that, for instance, A generates an analytic or, more generally, an eventually norm continuous semigroup. In particular, in this case the semigroup is exponentially stable if and only if s(A) := sup{Re λ : λ ∈ σ(A)} < 0. For this and related facts and the spectral theory of C_{0}–semigroups we refer to the monograph [20]. Unfortunately, in the non-autonomous situation such a criterion fails even for X = ℂ^{2}, [7, p.3]. In an appendix we present a rather drastic example illustrating this phenomenon for infinite dimensional X: There exist generators A(t), t ≥ 0, of uniformly bounded, commuting, positive C_{0}–semigroups on an L^{1}–space such that A(·) is smooth in a certain sense, (CP) is well-posed, and s(A(t)) = −∞ except for a sequence (t_{k}) where A(t_{k}) = 0. However, the corresponding evolution family U grows faster than any exponential. A related example for bounded operators A(t) is also constructed.

1991 Mathematics Subject Classification. Primary: 34G10; Secondary: 35K10, 47A55, 47D06.

Key words and phrases. Evolution equation, evolution family, exponential stability, exponential dichotomy, hyperbolicity, robustness, parabolic case, spectrum, evolution semigroup.

I thank Frank Räbiger and Rainer Nagel for many stimulating discussions. Part of this work was done during a visit at the University of Missouri–Columbia in spring 1997 where I experienced the kind hospitality of Yuri Latushkin and Timothy Randolph. Support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

Moreover, it seems to be rather difficult to find assumptions on unbounded operators A(t) which imply exponential dichotomy of (CP). In fact, to our knowledge the results in the literature are mostly restricted to exponential stability and to robustness of dichotomy, see the references below.

In our paper we present several sufficient conditions for exponential stability and dichotomy of (mostly parabolic) evolution equations. In Section 2 and 3 it is proved that exponential stability and dichotomy, respectively, of the semigroups (e^{τA(t)})_{τ≥0} with uniform constants imply the same properties for the solutions of (CP) provided that A(·) does not oscillate too much. Similar criteria for exponential stability can be found in, e.g., [7, 8] for bounded A(t) and in [2, 10, 12, 35] for parabolic equations, see also [21] and the references therein. Our results in Section 2 are consequences of a general approximation theorem for evolution families, Theorem 2.1. For exponential dichtomy, our main Theorem 3.7 is similar to known results for bounded operators A(t), [3, 7], and for delay equations, [18], but it seems to be the first theorem of this type for parabolic equations. We also refer to [19, §6.3] and the references therein concerning periodic parabolic equations, where the monodromy operator can be used. Further, in Section 4 we study robustness of exponential dichotomy. In view of the vast literature on this topic let us only mention [7, 8] for bounded A(t), [5, 15, 16] for bounded perturbations, and [6, 12, 17] for certain unbounded perturbations. We (partially) extend these results to time dependent Miyadera perturbations. In Example 2.4 and 3.8, our results are applied to parabolic partial differential equations.

Our approach is based on the use of evolution semigroups which we introduce now. Throughout let I ∈ {ℝ, ℝ_{+}}. Further, J ⊆ ℝ denotes a closed interval, and X is a Banach space. We set D_{J} = D := {(t, s) ∈ J^{2} : t ≥ s}. Let J′ := J \ inf J, where ℝ′ = ℝ. A family U = (U(t, s))_{(t,s)∈D} in the space L(X) of bounded linear operators on X is called evolution family if (E1) U(s, s) = Id, U(t, s) = U(t, r)U(r, s) for t ≥ r ≥ s in J and (E2) the mapping D ∋ (t, s) ↦ U(t, s) is strongly continuous. Notice that a C_{0}–semigroup (e^{tA})_{t≥0} yields an evolution family U by setting U(t, s) := e^{(t−s)A} for (t, s) ∈ D_{ℝ}. We say that an evolution family U on X solves the Cauchy problem (CP) on spaces Y_{s}, s ∈ J, (or that (CP) is well–posed) if Y_{s} is dense in X, U(t, s)Y_{s} ⊆ Y_{t} ⊆ D(A(t)) for (t, s) ∈ D, and u = U(·, s)x ∈ C^{1}(J ∩ (s, ∞), X) is the unique solution of (CP) for x ∈ Y_{s}. We remark that this concept of well–posedness could be formulated in terms of solutions of (CP) only, cf. [23].

The exponential growth bound ω(U) of an evolution family U is given by

ω(U) := inf{w ∈ ℝ : there is M_{w} ≥ 1 with ∥U(t, s)∥ ≤ M_{w} e^{w(t−s)} for (t, s) ∈ D}.

The evolution family U is called exponentially bounded if ω(U) < ∞, and U is exponentially stable if ω(U) < 0. Observe that ω(U) = −∞ if J is compact. Exponential dichotomy (or hyperbolicity) is introduced in Definition 3.1.

Let U be an exponentially bounded evolution family on X with index set D_{J}. Consider the Bochner–Lebesgue spaces L^{p}(J, X), 1 ≤ p < ∞, and the space C_{00}(J, X) of continuous functions f : J → X vanishing at inf J and at infinity (if J is unbounded) endowed with the sup–norm ∥ · ∥_{∞}. We write C_{0}(ℝ, X) instead of C_{00}(ℝ, X). On E = L^{p}(J, X) or C_{00}(J, X) we define by

(T (t)f) (s) := χ_{J}(s − t) U(s, s − t)f(s − t) = \{ \begin{matrix}U(s, s − t)f(s − t), & s, s − t ∈ J, \\ 0, & s ∈ J, s − t ∉ J, \end{matrix}

a bounded operator T (t) for t ≥ 0, where χ_{J} is the characteristic function of J. It is easily seen that T = (T (t))_{t≥0} is a strongly continuous semigroup on E, see e.g. [31], [33]. We call T the evolution semigroup induced by U and denote its generator by (G, D(G)). Concerning the theory of evolution semigroups we refer to [14, 15, 16, 22, 23, 24, 28, 29, 30, 31, 33] and the references therein. For U(t, s) ≡ Id the evolution semigroup coincides with the (right-) translation semigroup T_{0} = (T_{0}(t))_{t≥0} with generator G_{0}f = −f′. Notice that on C_{00}(J, X) we have D(G_{0}) = {f ∈ C^{1}(J, X) : f, f′ ∈ C_{00}(J, X)}. Finally, for operators (A(t), D(A(t))), t ∈ J, we define the multiplication operator Af := A(·)f(·) with D(A) := {f ∈ E : f(t) ∈ D(A(t)) (a.e.), A(·)f(·) ∈ E} on E = C_{00}(J, X), L^{p}(J, X). Using this notation, we state the following representation of G which can be shown as [15, Prop. 2.9] or [33, Prop. 1.13]. For simplicity, we set U(t, s) := 0 for t < s.

Lemma 1.1. Let U be an exponentially bounded evolution family on X and T be the induced evolution semigroup on E = C_{00}(J, X) or L^{p}(J, X), 1 ≤ p < ∞. Denote by D the linear span of functions of the form φ(·)U(·, s)x, where s ∈ J, φ ∈ C^{1}(J) with compact support contained in (s, ∞) ∩ J, and x ∈ Y_{s} for dense subspaces Y_{s} of X. Then T (t)D ⊆ D ⊆ D(G) and D is dense in E. If U solves (CP) on Y_{s}, then D ⊆ D(G_{0})∩D(A) and G is the closure of (G_{0} + A, D).

2. Exponential stability

In the sequel, we prove an approximation theorem for evolution families which implies several sufficient conditions for exponential stability of the solutions of (CP). We assume the well–posedness of the Cauchy problem (CP) given by linear operators A(t) on X which satisfy the following assumptions. (A) (A(t), D(A(t))) generates a C_{0}–semigroup (e^{τA(t)})_{τ≥0} on a Banach space X with

∥eτA(t) :DT