PFDE WITH NONAUTONOMOUS PAST

. We study Cauchy problems associated to partial diﬀerential equations with inﬁnite delay where the history function is modiﬁed by an evolution family. Using sophisticated tools from semigroup theory such as evolution semigroups, ex- trapolation spaces, or the critical spectrum, we prove well-posedness and characterize the asymptotic behavior of the solution semigroup by an operator-valued character- istic equation.


1.
Introduction. Linear partial differential equations with infinite delay can be written in abstract form aṡ x 0 = f.
Here, the function x(·) takes values in a Banach space X, B is a linear operator on X, the history function x t : R − → X is defined by We refer to the monographs [5], [8] or [35] for the general theory, and to [6], [30] or [32] for some recent contributions to the theory of delay semigroups.
Our approach in this paper to more general delay equations (see (1) and (2) below) is inspired by Section VI.6 of [7], where it is shown that (DDE) is well-posed if and only 954 SIMON BRENDLE AND RAINER NAGEL if (ACP) is well-posed in the semigroup sense. Moreover, if B generates a strongly continuous semigroup (e tB ) t≥0 and if Φ ∈ L(E, X), then G B,Φ generates a strongly continuous semigroup (T B,Φ (t)) t≥0 on E (cf. [7], Theorem VI.6.1), hence (ACP) is well-posed. This delay semigroup satisfies the so-called translation property (T B,Φ (t)f )(s) = f (s + t) for s + t ≤ 0 (T B,Φ (s + t)f )(0) for s + t ≥ 0 for all f ∈ E (cf. [7], Lemma VI.6.2), which implies that the solutions of (DDE) are given by for t ≤ 0 (T B,Φ (t)f )(0) for t ≥ 0 (cf. [7], Corollary VI.6.3). Moreover, the translation property says that the delay semigroup (T B,Φ (t)) t≥0 acts on a history function f ∈ E by left translation and that, in particular, the value f (0) at time 0 remains unchanged while being translated into the past. Assume now that, while time is passing, a backward evolution family (cf. [4] p. 8 or [25]) modifies the values of the history function. More precisely, assume that (U (s, r)) r≤s≤0 is a family of bounded linear operators on X satisfying and that the semigroup (T B,Φ (t)) t≥0 solving (DDE) satisfies a modified translation property of the form It is our goal in this paper to study the combination of these two equations. For finite dimensional X similar equations occur, e.g., in [22]. For unbounded A(s) however, no systematic study is known to us. However, the population equation with diffusion studied by Nickel, Rhandi and Schnaubelt in [24], [28], [29] can be viewed as an equation of this type.
In a first step, we show that, under appropriate assumptions, a strongly continuous semigroup (T B,Φ (t)) t≥0 exists on E such that u(t, s) := (T B,Φ (t)f )(s) solves (1) and (2) in a mild sense for an initial value f ∈ E. In a next step, we characterize the spectrum of the generator G B,Φ of (T B,Φ (t)) t≥0 by a characteristic operator equation in X. Finally, we estimate the critical growth bound of (T B,Φ (t)) t≥0 and obtain as a consequence that the spectrum of G B,Φ determines the growth bound of (T B,Φ (t)) t≥0 , hence of the solutions of (1) and (2). By adding appropriate positivity assumptions, the stability criteria become much simpler and lead, in the case of the example discussed in Section 5, to quite explicit results.
2. Well-Posedness. We start by stating the assumptions which we will keep throughout the paper.
Assumption 2.1. Let B be the generator of a strongly continuous semigroup (e tB ) t≥0 on a Banach space X. On E := C 0 (R − , X) we consider a bounded linear operator Φ from E into X. Finally, let U = (U (s, r)) r≤s≤0 be a strongly continuous backward evolution family, i.e., a family of bounded linear operators on X satisfying and such that the mapping (s, r) → U (s, r) is strongly continuous. Moreover, we assume that (U (s, r)) r≤s≤0 is exponentially bounded, i.e., there exist constants M and w ∈ R such that U (s, r) ≤ M e w(s−r) for all r ≤ s ≤ 0. We then define the growth bound of (U (s, r)) r≤s≤0 by In a first step we extend (U (s, r)) r≤s≤0 to a backward evolution family (Ũ (s, r)) r≤s on all of R and then define a corresponding evolution semigroup. Such evolution semigroups have been studied systematically in the monograph [3] or in the survey article by Schnaubelt in [7], Section VI.9, and have been applied with great success to characterize qualitative properties of solutions to nonautonomous evolution equations (cf. [19], [20], [31]). Definition 2.2. (i) The backward evolution family U = (U (s, r)) r≤s≤0 on X is extended to a backward evolution familyŨ = (Ũ (s, r)) r≤s bỹ (ii) OnẼ := C 0 (R, X) we then define the corresponding evolution semigroupT = (T (t)) t≥0 by for all s ∈ R, t ≥ 0, andf ∈Ẽ.
Then one easily proves the following property of (T (t)) t≥0 . Proposition 2.3. With the above definitions, (T (t)) t≥0 is a strongly continuous semigroup onẼ, and its generatorG is a local operator in the sense thatf ∈ D(G), f (s) = 0 for all a < s < b impliesGf (s) = 0 for all a < s < b.
Remark that we did not assume (Ũ (s, r)) r≤s to be differentiable, and hence the precise description ofG and its domain D(G) is difficult. However, assuming differentiability, the generator of an evolution semigroup has been described more explicitly in [23], Proposition 2.7. We adapt this result to our situation.
We remark that, in general, the operatorsÃ(s) may have trivial domain. On the other hand, if the evolution family corresponds to a well-posed Cauchy problem, then there is a core D of D(G) such thatGf =f +Ã(·)f forf ∈ D (cf. [3], Theorem 3.12 or [25], Theorem 2.3).
For the study of our delay differential equation, we have to restrict the evolution semigroup (T (t)) t≥0 to the space E = C 0 (R − , X), and therefore need a boundary condition for the left shift. Such evolution semigroups on the half line have been considered recently in, e.g., [11] or [12] in order to study the asymptotic behavior of the corresponding nonautonomous problems. Here, we use the semigroup (e tB ) t≥0 generated by B to define the appropriate evolution semigroup.
Then one can again prove the semigroup property of (T B,0 (t)) t≥0 .
We denote its generator by G B,0 and try to express it in terms of the generatorG of (T (t)) t≥0 . To this purpose we first restrictG to the space E.
SinceG is a local operator (cf. Proposition 2.3), G is well-defined, and we will obtain the generator G B,0 of (T B,0 (t)) t≥0 as a restriction of G.
Proposition 2.8. The generator G B,0 of (T B,0 (t)) t≥0 is given by , we can extend f to a functionf ∈ D(G). We then have . Choose now a continuously differentiable function g : R + → X with compact support such that g(0) = f (0) and g (0) = Bf (0). Then, the functionf defined byf We now use the delay operator Φ ∈ L(E, X) to define the operator In order to show that this operator is again a generator we use a technique introduced by Rhandi [27] and extend G B,Φ to an operator on E := X × E.
Lemma 2.9. The operator is a Hille-Yosida operator on the Banach space E := X × E.
Using the operator e λ , the resolvent of G B,0 can be expressed as for Re λ > ω 0 (T B,0 ). In order to show this we take ( x f ) ∈ E and put g := e λ R(λ, B)x + R(λ, G B,0 )f. We now prove that the semigroup T B,Φ = (T B,Φ (t)) t≥0 generated by G B,Φ satisfies a modified translation property. This will be deduced from the following property of the operator G.
Proof. Since f ∈ D(G), we can extend f to a functionf on R − contained in D(G). By the definition (see Definition 2.2) of (T (t)) t≥0 , we havẽ
Lemma 2.13. The semigroup (T B,Φ (t)) t≥0 satisfies Proof. It suffices to prove the assertion for f ∈ D(G B,Φ ). Since G B,Φ is a restriction of G, we know from Lemma 2.12 that can be written as a function of r and s + t. From this it follows that Proposition 2.14. The semigroup (T B,Φ (t)) t≥0 satisfies Proof. It suffices to prove the assertion for f ∈ D(G B,Φ ). In this case, we have Therefore, we obtain Using the translation property from Lemma 2.13, the assertion follows.
Remark 2.15. If the terms in the above formula for (T B,Φ (t)f )(s) are all differentiable and belong to the domains of the relevant operators, we obtain a (classical) solution of the equations (1) and (2) (see [9] for details). Therefore, for each f ∈ E, the function u(t, s) := (T B,Φ (t)f )(s) can be regarded as a mild solution of (1) and (2).
3. Spectral Theory. We are now trying to determine the spectrum σ(G B,Φ ) of the generator G B,Φ by a condition in the space X instead of E = C 0 (R − , X). To that purpose, we recall that, for Re λ > ω 0 (T B,0 ), we defined in the proof of Lemma 2.9 operators e λ ∈ L(X, E) by (e λ x)(s) := e λs U (0, s)x for s ≤ 0 and x ∈ X. Since Φ ∈ L(E, X), we obtain that Φe λ ∈ L(X, X).
Proof. It follows from [7], Theorem II.5.15 that the spectra of G B,Φ and G B,Φ coincide. Hence, it remains to show that To this end, we write The above condition is the appropriate analogue of the characteristic equation for delay differential equations in finite dimensions (see e.g. [8], Lemma 7.2.1) and is a characteristic operator equation as studied in [14], [15]. It is challenging to use it for the qualitative study of nonlinear problems ( [10] or [26]). However, the determination of all λ such that λ ∈ σ(B + Φe λ ) remains a very difficult task. In the remaining part of this section we show how positivity arguments in combination with Theorem 3.1 yield a useful estimate for the spectral bound s(G B,Φ ) of the generator G B,Φ .
To this end, we assume X to be a Banach lattice. Then E becomes a Banach lattice as well. Furthermore, we assume that the semigroup (e tB ) t≥0 generated by B and the delay operator Φ are both positive. Finally, we assume that the backward evolution family (U (s, r)) r≤s≤0 consists of positive operators. For the general theory of positive semigroups we refer to [13], Chap VI.1.b in [7] or [34]. Proof. We imitate the proof of [7], Lemma VI.6.15. In the first part of the proof, we show that s(B + Φe λ ) < λ implies s(G B,Φ ) < λ. In the second part, we show that s(B + Φe λ ) ≥ λ implies s(G B,Φ ) ≥ λ.
For a Banach space E and a strongly continuous semigroup T = (T (t)) t≥0 on E, we consider the Banach spaceẼ := ∞ (E) of all bounded sequences in E. We extend the semigroup (T (t)) t≥0 to this space and obtain a new semigroupT = (T (t)) t≥0 byT (t)f := (T (t)f n ) n∈N forf = (f n ) n∈N .
For this extended semigroup we consider its space of strong continuitỹ This subspace is closed and (T (t)) t≥0 -invariant. Therefore, the semigroup (T (t)) t≥0 induces a quotient semigroupT = (T (t)) t≥0 on the quotient spaceÊ :=Ẽ/Ẽ T , which is given byT The critical spectrum of T (t) is then defined as σ crit (T (t)) := σ(T (t)), while the critical growth bound of (T (t)) t≥0 is defined as ω crit (T ) := inf{w ∈ R : ∃M such that T (t) ≤ M e wt for all t ≥ 0}.
Proposition 4.1. For a strongly continuous semigroup T = (T (t)) t≥0 , the critical growth bound is given by As the main property of the critical spectrum, we state the following partial spectral mapping theorem and, as a consequence, a characterization of the growth bound ω 0 (T ) (see [17], Theorem 3.2).
Theorem 4.2. For a strongly continuous semigroup T = (T (t)) t≥0 with generator G the following statements hold.
We now apply these concepts to our semigroup (T B,Φ (t)) t≥0 and use the fact that the critical growth bound remains unchanged if the new semigroup differs from the given semigroup by a norm continuous operator family (see [2], Section 4).
Theorem 4.3. If the semigroup (e tB ) t≥0 generated by B is immediately norm continuous, then the growth bound of (T B,Φ (t)) t≥0 is given by Proof. By Corollary 2.14, we have for all f ∈ E. Since the semigroup (e tB ) t≥0 is immediately norm continuous, it follows that the operator family (T B,Φ (t) − T 0,Φ (t)) t≥0 is norm continuous. By Proposition 4.1, the critical growth bounds of (T B,Φ (t)) t≥0 and (T B,0 (t)) t≥0 coincide, i.e. ω crit (T B,Φ ) = ω crit (T B,0 ). We now determine the critical growth bound of (T B,0 (t)) t≥0 . From the definition of (T B,0 (t)) t≥0 we obtain the estimates lim h↓0 T B,0 (t + h) − T B,0 (t) ≥ sup Therefore, we conclude that and, by Theorem 4.2, the growth bound of (T B,Φ (t)) t≥0 becomes

Example.
Let Ω be a bounded smooth domain in R n . The Dirichlet Laplacian generates an analytic semigroup (e t∆ ) t≥0 on X := L 2 (Ω). We then define operators A(s) as A(s) := a(s)∆, where the function 0 ≤ a(·) ∈ L 1 loc (R − ). These operators generate a backward evolution family (U (s, r)) r≤s≤0 given by U (s, r) = e ( s r a(σ) dσ)∆ for r ≤ s ≤ 0. We then have U (s, r) = e ( s r a(σ) dσ)λ0 , where λ 0 denotes the largest eigenvalue of ∆. This identity allows us to compute directly the growth bound of (U (s, r)) r≤s≤0 .
Proposition 5.1. The growth bound of (U (s, r)) r≤s≤0 is given by for f ∈ E, where 0 ≤ ϕ(·) ∈ L 1 (R − ). The semigroup (T B,Φ (t)) t≥0 from Section 2 is then well defined and solves the equations (1) and (2) corresponding to Φ and A(·). In order to compute the growth bound of (T B,Φ (t)) t≥0 , it suffices to compute the spectral bound of its generator.