Notes on spectrum and exponential decay in nonautonomous evolutionary equations

2016, No.19, 1–15; doi: 10.14232/ejqtde.2016.8.19 http://www.math.u-szeged.hu/ejqtde/

Notes on spectrum and exponential decay in nonautonomous evolutionary equations

Christian Pötzsche

and Evamaria Russ

Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt, Austria

Appeared 11 August 2016 Communicated by Tibor Krisztin

Abstract. We first determine the dichotomy (Sacker–Sell) spectrum for certain nonau- tonomous linear evolutionary equations induced by a class of parabolic PDE systems.

Having this information at hand, we underline the applicability of our second result:

If the widths of the gaps in the dichotomy spectrum are bounded away from 0, then one can rule out the existence of super-exponentially decaying (i.e. slow) solutions of semi-linear evolutionary equations.

Keywords: nonautonomous evolutionary equation, dichotomy spectrum, Sacker–Sell spectrum, slow solutions.

2010 Mathematics Subject Classification: 37C60, 35K58, 35K91.

1 Motivation

When investigating evolutionary equations near non-constant reference solutions, in the vicin- ity of compact invariant sets (e.g. nontrivial attractors, homo- or heteroclinic solutions) or under time-variant parameters, one is confronted with a nonautonomous problem: the vari- ational equation becomes explicitly time-dependent and an appropriate spectral theory turns out indispensable in order to determine e.g. linear stability. Due to its ambient robustness properties, uniform asymptotic stability is a favorable concept and can be determined by means of the dichotomy (dynamical or Sacker–Sell) spectrum (cf. [3,14,16]). Actually, appli- cations of the aforesaid dichotomy spectrum Σ ⊆ ^R reach further than basic stability issues.

For instance gaps in Σ allow to construct the entire hierarchy of stable and unstable man- ifolds, as well as their invariant foliations. Under particular assumptions on the spectrum it is even possible to extend Lu’s topological linearization result [11] to a class of nonau- tonomous evolutionary equations in Banach spaces. Yet, specifically this endeavor requests certain preparations concerning the dichotomy spectrum.

First, for the sake of relevant examples,Σhas to be known (at least qualitatively) in various types of evolutionary differential equations. For instance, delay differential equations were considered in [12]. Building on previous results from [4,9,10], in our Section3 we determine

BCorresponding author. Email: christian.poetzsche@aau.at

the dichotomy spectrum for linear evolutionary equations whose infinitesimal generator is sectorial with compact resolvent. Canonical examples include uniformly elliptic differential operators or the poly-Laplacian under the standard boundary conditions.

Second, extending the topological linearization argument of [11] requires evolutionary equations without nontrivial small solutions. This class of functions decays to 0 faster than any exponential function and typically occurs for delay differential equations (cf. [6, pp. 74ff]).

Parabolic PDEs, on the other hand, cannot have slow solutions and [1, Lemma 5] serves as standard reference. In Section 4 we generalize this technical, but helpful and interesting result to semi-linear equations and allow a time-dependent linear part; furthermore, our proof is slightly simpler. This necessitates to impose two central assumptions: (a) The invariant projectors associated to the dichotomy spectrum are complete. In the autonomous special case this means that the infinitesimal generator has a complete set of eigenvectors. (b) Moreover, the width of the gaps inΣneeds to be uniformly bounded away from 0.

Indeed, the note at hand is essentially a supplement to [13] and provides preparations being crucial there. Nonetheless, we feel the present examples and results are both handy and of independent interest when dealing with nonautonomous parabolic PDEs, their geometric theory and beyond. Our approach to nonautonomous dynamics is via evolution families and 2-parameter semigroups, rather than skew-product semiflows as used in [4,9,10]. We feel this is more appropriate in the present situation since one can omit e.g. uniform continuity properties of the coefficient functions (in order to guarantee a compact base space). Finally, compared to [4,9,10] more general time-dependencies and a wider flexibility on the differential operator is allowed.

Notation. The kernel of a linear operator Aon a Banach space X is denoted by N(A), R(A) is its range and id_X the identity. We write σ(A) for the spectrum and σ_p(A) for the point spectrum of A. The Kronecker symbol is denoted asδ_kl ∈ {^{0, 1}}^,k,l∈ ^N.

Given nonempty subsetsB,C⊆^Randλ∈^Rit is convenient to abbreviate λ+B:={^λ+b∈^{R: b}∈ ^B}^{, B}+C:={^b+c∈^{R: b}∈^B,c∈^C}^.

2 Evolution families, dichotomies and Bohl exponents

For an unbounded interval J ⊆ ^R and a Banach space (X,k·k), let us introduce our central notions: We begin with a useful generalization of the semigroup concept when dealing with time-dependent problems: An evolution family T : {(t,s)∈ ^J×^{J : s}≤^t} → ^L(X) on X is defined as a mapping such that(t,s)7→^T(t,s)uis continuous for allu∈X, which furthermore fulfills

• T(t,t) =id_XandT(t,s)_T(s,τ) =_T(t,τ)for all τ≤^s≤^t

• there exist realsK₀ ≥^1,α0∈ ^{Rsuch that}k^T(t,s)k ≤^K0e^α0(t−s)for alls ≤^t.

One says the evolution familyThas anexponential dichotomy(ED for short) on J, if there exists a projectorP: J →^L(_X)and realsK≥^1,α>0 such that

• T(t,s)_P(s) =_P(t)_T(t,s)for alls≤ ^t(Pis aninvariant projector)

• ¯T(t,s):=T(t,s)|N(_P(s)) :N(P(s))→ ^N(P(t))is an isomorphism fors <t

• kT(t,s)_P(s)k ≤^{Ke−α(t−s)and}kT^¯(s,t) [id_X−P(t)]k ≤^{Keα(s−t)fors} ≤^t.

LetN⊆^Nbe an index set. A family of projectorsP_k : J → ^L(X),k∈ N, is calledcomplete, if one has

u=

∑

k∈^N

P_k(τ)u for all(τ,u)∈ ^J×X.

With γ ∈ ^{R we write T}γ(t,s) := e^γ(s−t)T(t,s) for the associated scaled evolution family. The dichotomy spectrumΣJ ofTis

ΣJ :={^γ∈^R: Tγ admits no ED on J}^.

For evolution families as defined above, ΣJ is a closed subset of (−∞,α₀]. In general, the spectrum depends on the interval J and at any rate it holds ΣJ ⊆ ΣR. If the evolution family Tis the evolution operator of an abstract nonautonomous differential equation

˙

u =A(t)u, (L)

then we writeΣJ(A)for the dichotomy spectrum. By definition, one has the relation

ΣJ(A+λ) =λ+ΣJ(A) for allλ∈^{R. (2.1)} As a prototype example let us constitute

Example 2.1(dichotomy spectrum in finite dimensions). In caseX=Rⁿ, a linear ODE

˙

u =B(t)u (2.2)

with continuous coefficient operatorB : J → ^L(Rⁿ)and bounded growth generates an evolu- tion familyT(t,s)∈ ^L(R^d),t,s ∈ J. Its dichotomy spectrum has the form

ΣJ(B) =

m

[

j=1

hλ⁻_j ,λ⁺_j i ,

where the reals λ⁻_j ,λ⁺_j are ordered according to λ⁻_m ≤ ^λ+m < · · · < _λ⁻₁ ≤ ^λ+₁. Each of the m≤n spectral intervals[λ⁻_j ,λ⁺_j ]corresponds to an invariant vector bundle

Xj = (t,x)∈ ^J×^{Rn: x}∈ ^R(p_j(t)) for 1≤ ^j≤ ^m,

where p_j : J → ^L(Rⁿ)is an invariant projector and one has the Whitney sum (cf. [14,16]) J×^Rd =X1⊕ · · · ⊕ Xm.

For the further special case of scalar ODEs ˙u = a(t)u the dichotomy spectrum allows an explicit representation. Thereto, given a continuous a : J → ^{R, itsupper Bohl resp.lower Bohl} exponentis defined as

β_J(a):=lim sup

T→∞ sup

τ∈J [τ,τ+T]⊆^J

1 T

Z _τ+T

τ a, β_J(a):=lim inf

T→∞ inf

τ∈^J [τ,τ+T]⊆^J

1 T

Z _τ+T

τ a.

One obviously hasβ_J(a)≤ ^β_J(a)and the integrability conditions sup

0≤^t−^s≤¹ Z _t

s a< _∞, sup

0≤^s−^t≤¹ Z _t

s a <_∞ (2.3)

are necessary and sufficient for finite Bohl exponents, i.e. β_J(a)<_{∞ resp.} −^∞ < _β

J(a) (for this, see [7, p. 259, Proposition 3.3.14]). The boundednessγ:=sup_t∈J|^a(t)|<_∞even guaran- tees that−γ≤ β_J(a)≤β_J(a)≤ γ. Moreover, Bohl exponents satisfy

β_J(λ+a) =λ+β_J(a), β_J(λ+a) =λ+β_J(a) for all λ∈^R.

Example 2.2. (1) Ifa(t)≡^{α, then} β_J(a) =β_J(a) =αfor allα∈^R.

(2) If a : J → ^{R is} θ-periodic with some θ > _{0, i.e. a}(t+θ) = a(t) holds for all t ∈ ^J satisfyingt+θ∈ J, then Bohl exponents are the means

β_J(a) =β_J(a) = ¹ θ

Z _t+θ

t a(r)dr for allt ∈ ^J.

(3) If a(t) = ^α+₂^β + ^β−₂^αsin lnt with reals α ≤ ^{β, then β}_J(a) =α and β_J(a) = β holds for every unbounded subintervalJ ⊆(0,∞).

(4) Ifa:R→^Ris continuous and fulfills lim_t→±∞a(t) =α^± for realsα⁻,α⁺, then β_(−∞,τ](a) =β_(−∞,τ](a) =α⁻, β_[τ,∞)(a) =β_[τ,∞)(a) =α⁺ for allτ∈^R,

β_R(a) =min

α⁻,α⁺ , β_R(a) =max

α⁻,α⁺ .

Equations with such asymptotically constant or periodic coefficients were studied in [10].

The importance of Bohl exponents is due to their role in stability theory and as boundary points of the dichotomy spectrum. Our above Example 2.2 can be used in the following lemma.

Lemma 2.3. If a continuous function a : J → ^R fulfills the integrability conditions (2.3), then the ordinary differential equation

u˙ = a(t)u (2.4)

inXpossesses the dichotomy spectrumΣJ(a) = [β_J(a),β_J(a)]. Proof. Since (2.4) has the evolution operatorT(t,s) = expRt

s a

id_X for all t,s ∈ J, the claim follows from [8, Proposition A.2].

3 Dichotomy spectrum of parabolic equations

LetXbe a separable Hilbert space equipped with the inner producth·^,·i^. 3.1 Generators with discrete spectrum

Let us supposeA: D(A)⊆^X→^Xis a linear unbounded operator generating aC₀-semigroup S:[0,∞)→^L(X)and

• σ(A) =σp(A) ={^λk : k∈ ^N} ⊆^Rwith

λ1≥^λ2 ≥ · · ·^,

where every eigenvalueλk is repeated as many times as its finitemultiplicitygiven by µ_k =dimN(A−^λkid_X)

• each corresponding eigenspace

N(A−^λkid_X) =spann

e^k₁, . . . ,e^k_µko is spanned by orthogonal eigenvectorse^k₁, . . . ,e^k_µk ∈ ^{Xof A. Thus,}

Πk :=

µ_k

j

∑

• n

e^k_j ∈ ^{X: 1}≤^j≤µk andk∈^Nois a complete orthonormal set in X.

The typical examples for such operators are as follows, where Ω ⊆ ^Rd denotes a bounded domain with piecewise smooth boundary.

Example 3.1(uniformly elliptic differential operators). Consider a uniformly elliptic differen- tial operator in divergence form (see e.g. [5, pp. 354ff.])

Lu(x) =

∑

D_j a^ij(x)D_iu(x) for allx ∈^Ω

with coefficientsa^ij ∈^C∞(Ω^¯)satisfyinga^ij = a^ji for all 1≤^i,j≤d. If we define the operator (Au)(x) =Lu(x)

on X= L²(Ω), then the above properties hold:

(1) In order to capture Dirichlet boundary conditionsu(x)≡^{0 on} ∂Ω, choose D(A) =H²(Ω)∩^H10(Ω).

The principle eigenvalue λ1<0 is negative; the eigenfunctions are contained in H₀¹(Ω). (2) Concerning Neumann boundary conditionsD_νu(x)≡^{0 on} ∂Ω, choose the domain

D(A) =u∈ ^H2(Ω): Dνu(x)≡^{0 on}∂Ω . For the Laplacian one hasλ₁ =0.

In particular, ifLis the Laplacian∆equipped with Dirichlet, Neumann or Robin boundary conditions (i.e. au(x) +bD_νu(x) ≡ ^{0 on} ∂Ω), then according to Weyl’s Law the eigenvalues behave asymptotically asλ_k ∼^Cd(_Ω)k^2/d fork →∞.

Example 3.2(poly-Laplacian). Givenm∈^Nlet us consider the poly-Laplacian Lu(x) =−(−^∆)^mu(x) for all x∈^Ω

with homogeneous boundary conditions u(x) ≡ ^Dνu(x) ≡ · · · ≡ ^Dν^m−1u(x) ≡ ^{0 on} ∂Ω. It yields an operator

(Au)(x):=Lu(x)

on X = L²(Ω)with D(A) = H^2m(Ω)∩^H₀^m(Ω) fulfilling our above properties. The principle eigenvalue isλ1<0 and thanks to [2, p. 12, Theorem 1.11], the eigenvalues behave asymptot- ically asλ_k ∼^Cd(_Ω)k^2m/d fork→∞.

Example 3.3(Laplacian in 1d). The special cases withΩ= (0,`),` >_{0, and} Lu(x) =uxx(x) for all 0<_x < `

allow more explicit results:

(1) For Dirichlet boundary conditions,

D(A) = H²(0,`)∩<sup>H</sup>0<sup>1</sup>(0,`)

yields simple eigenvaluesλ_k =− ^πk` )² with eigenfunctionse^k(x) = q2

`sin <sup>πk</sup><sub>`</sub> x

fork∈^N.

(2) For Neumann boundary conditions,

D(A) =u∈ ^H2(0,`): u<sub>x</sub>(0) =u<sub>x</sub>(`) =0 , all eigenvaluesλk =− ^π(k_`⁻¹⁾² are simple with the eigenfunctions

e¹(x)≡ √¹

`^{, e}

k(x) = r2

` ^cos

π(k−¹)

` ^x

for allk ≥^2.

Finally, in the above examples−^A:D(A)⊆ ^X→^Xis a sectorial operator andAgenerates an analytic semigroup S : [0,∞) → ^L(X) on X (cf. [15, p. 106, Theorem 38.1]) allowing the Fourier representation

S(t)u=

∑

k∈^N

e^λktΠku for all t≥^{0, u}∈ ^{X. (3.2)}

3.2 Systems of parabolic equations

Let L denote a differential operator from the previous Examples 3.1–3.3. Consider the n- dimensional system of PDEs







u¹_t =d₁₁Lu¹+· · ·+d_1nLuⁿ,

uⁿ_t =... d_n1Lu¹+· · ·+dnnLuⁿ, (3.3) which briefly can be written as

U_t =DLU

with U = (u¹, . . . ,uⁿ)^T, LU := (Lu¹, . . . ,Luⁿ)^T. The “diffusion matrix” D ∈ ^L(Rⁿ) has the entriesd_ij and is supposed to be symmetric positive-definite.

In order to formulate (3.3) as an abstract evolutionary equation in a separable Hilbert space, we equip the cartesian productX:= Xⁿ with the inner product

hh^U,Vii^:=

∑

j=1h^uj,vji ^{for allU,V} ∈X.

Endowing the PDE system (3.3) with ambient boundary conditions allows us to define (AU)(x):=DLU(x) for all x∈Ω (3.4) as an operator on X = L²(Ω)ⁿ and the domain D(A) = D(A)ⁿ. The diagonalizability as- sumption onDshows that also −Ais sectorial and thusAgenerates an analytical semigroup S:[0,∞)→ ^L(X)onX. Thanks to (3.2) it allows the Fourier representation

S(t)U=

∑

k∈^N

e^λktDP_kU (3.5)

withU∈ ^Xand the complete family(P_k)_k∈Nof orthogonal projections on Xgiven by P_k =diag(_Πk, . . . ,Πk).

Here,Πk ∈ ^L(X)are the orthogonal projections from (3.1) and one has

P_kP_l =δ_klP_l for allk,l∈^{N. (3.6)} Lemma 3.4. Under the above assumptions it isS(t)P_k =P_kS(t)for all t≥^{0, k}∈^N.

Proof. BecauseDis a positive-definite matrix, there exists an invertibleS∈ ^L(Rⁿ)such that SDS⁻¹=diag(d₁, . . . ,dn)

with eigenvalues d_j > 0 for all 1≤ ^j≤ n. Suppose that the entries of S⁻¹ are denoted by ˜s_ij. GivenU∈^Xwe first obtain

P_jS⁻¹U=P_j

∑

l=1







˜ s_1lu^l

...

˜ s_nlu^l





=

∑







˜ s_1lΠju^l

...

˜ s_nlΠju^l





= S⁻¹





 Πju¹

...

Πjuⁿ





U=S⁻¹P_jU (3.7) for all j∈^Nand this implies

S(P_jS(t)U)^(3.5)= S

P_j

∑

k∈^N

e^{λktS−1diag(d1,...,dn)S}P_kU

= S

P_jS⁻¹

∑

k∈^N

e^{λktdiag(d1,...,dn)}SP_kU

(3.7)

=

∑

k∈^N

P_jP_k





 e^λktd1

...

e^λktdn







∑







s_1lΠku^l ...

s_nlΠ_ku^l







(3.6)

= P_j

∑

l=1







e^λjtd1s_1lΠju^l ...

e^λjtdns_nlΠju^l





 =

∑







e^tλjd1s_1lΠju^l ...

e^tλjdns_nlΠju^l







= e^{tλjdiag(d1,...,dn)}SP_jU=Se^tλjDP_jU^(3.6)= S

∑

k∈^N

e^λktDP_kP_jU

(3.5)

= S(S(t)P_jU) for all t≥^0.

Thus, the claim is established by multiplying withS⁻¹from the left.

We first capture the effect of a scalar multiplicative and time-dependent perturbation on the dichotomy spectrum of (3.3). Thereto, assume thata: J →(0,∞)is a continuous function fulfilling the integrability conditions (2.3). Endowed with ambient boundary conditions the system of parabolic equations

U_t= a(t)DLU

can be formulated as nonautonomous abstract evolutionary equation

u˙ =a(t)_Au, (3.8)

whose evolution operatorT(t,s)∈^L(X)is given by T(t,s) =S

_{Z t}

s a (3.5)

=

∑

k∈^N

e^λkRstaDP_k for all s≤^{t. (3.9)} This representation allows us to obtain

Theorem 3.5(multiplicative perturbation). The dichotomy spectrum of the evolutionary equation (3.8)is

ΣJ(_aA) = ^[

k∈^N n

[

j=1

h

β_J(λ_kd_ja),β_J(λ_kd_ja)ⁱ

withσ(D) ={^d1, . . . ,dn}and complete invariant projectors P_k : J →^L(_X), k∈^N.

Proof. Thanks to Lemma3.4and (3.9) we obtain for everyk∈ ^Nthat P_kT(t,s) =P_kS

Z _t

s a

=S Z _t

s a

P_k =T(t,s)P_k for alls≤ ^t.

Hence, the finite-dimensional vector bundlesXk :={(t,U)∈ ^J×X: U ∈^R(P_k)}are invariant w.r.t. (3.8). Thanks to (3.9), inside of eachXk the dynamics is determined by

u˙ =λ_ka(t)Du, (3.10)

having an evolution operator satisfyingT^k(t,s):=T(t,s)P_k. It consequently follows T(t,s) =

∑

k∈^N

T^k(t,s) for all s≤^t and thus

ΣJ(_aA) = ^[

k∈^N

ΣJ(aλ_kD).

BecauseDis assumed to be symmetric, the ODEs (3.10) are kinematically similar to the diag- onal systems ˙u = λ_ka(t)diag(d1, . . . ,dn)u for allk ∈ N. Since kinematic similarity leaves the dichotomy spectrum invariant, one obtains

ΣJ(aλ_kD) =ΣJ(aλ_kdiag(d1, . . . ,dn)) =

n

[

l=1

ΣJ(aλ_kd_l) for all k∈^N

due to the fact that the spectrum of diagonal systems is the union of their diagonal spectra.

Then the assertion follows with Lemma2.3.

Example 3.6(periodic case). If a: J →(0,∞)isθ-periodic, then Example2.2(2) guarantees ΣJ(aA) = ¹

θ Z _t+θ

t a

n

[

j=1

d_j ^[

k∈^N

{^λk}^,

i.e. one obtains a discrete spectrum preserving the asymptotics ofλ_kfork→∞, provided that the meanRt+θ

t a6=0 does not vanish.

Example 3.7(asymptotically autonomous case). Letσ(D) ={¹}and suppose the eigenvalues ofLform a strictly decreasing sequence in(−∞, 0]with

klim→∞λ_k =−∞, lim

k→∞

λ_k+1

λk =1.

For a : R → (0,∞) satisfying the limit relations lim_t→±∞a(t) = α^± with 0 < _α⁻,α⁺ we set µ:= min{^α+,α−}^,µ:=max{^α−,α+}and obtain from Theorem3.5that

ΣR(_aA) = ^[

k∈^N

h

λ_kµ,λ_kµi .

If α⁺ = α⁻, then ΣR(aA) = α^+S_k∈N{^λk}. Otherwise, since the sequence ^λ_λ^k+_k¹

k∈^N ap- proaches its limit from above, there is a minimal k^∗ ∈ ^{Nwith λ}k+1/λ_k < _{µ/µfor allk} ≥ ^k∗. We derive µλ_k+1 > _µλk and hence successive spectral intervals

λ_kµ,λ_kµ

overlap for every k≥^k∗. Thus, the dichotomy spectrum consists only of finitely many intervals

ΣR(aA) =−^∞,λk^∗µi

∪

k^∗−¹ [

k=1

h

λ_kµ,λ_kµi .

Our second aim is to describe the impact of linear-homogeneous perturbations in (3.3).

Given a continuous matrix-valued function B: J → ^L(Rⁿ)we consider the PDEs U_t = a(t)LU+B(t)U.

After fixing ambient boundary conditions, it gives rise to the abstract nonautonomous evolu- tionary equation

˙

u=a(t)A+B(t)u (3.11)

with (AU)(x):=LU(x)and(B(t)U)(x):= B(t)U(x)for allt ∈ ^J,U∈Xandx∈Ω.

Theorem 3.8 (homogeneous perturbation). The dichotomy spectrum of the evolutionary equation (3.11)is

ΣJ(aA+B) = ^[

k∈^N

ΣJ(λka+B)

and possesses complete projectors p_l(·)P_k : J →^L(X),1 ≤^l ≤^{m, k} ∈^{N, where p}j : J → ^L(R^d)are the invariant projectors from Example2.1.

Proof. We subdivide the proof into two steps.

(I) Fort ∈ ^{J we set}P^l_k(t):= p_l(t)P_k ∈ ^L(_X)and write p^l_ij for the components of p_l. Because the orthogonal projections Πk ∈^L(X)are linear mappings, we obtain

P^l_k(t)U = p_l(t)





 Πku¹

...

Πkuⁿ





=

∑









p^l_1j(t)Πku¹ ...

p^l_nj(t)_Πkuⁿ









= P_k

∑

j=1









p^l_1j(t)u¹ ...

p^l_nj(t)uⁿ









=P_kp_l(t)U

for allU= (u¹, . . . ,uⁿ)∈^{Xand thus}

P^l_k(t) =P_kp_l(t). (3.12)

SinceP_k is a projection and p_l a projector, this allows us to show

P^l_k(t)²= p_l(t)P_kp_l(t)P_k ^(3.12)= p_l(t)P_kp_l(t)^(3.12)= P^l_k(t) and thereforep_l(·)P_k : J → ^L(_X)is a projector for all k∈^{N, 1}≤^l≤^m.

(II) LetTB(t,s)∈ ^L(X),s,t∈ ^J, denote the evolution family generated by the ODE u˙ =_B(t)u

in X. If the evolution family T_B(t,s) ∈ ^L(Rⁿ) of (2.2) has the components t_ij(t,s) ∈ ^R, 1≤^i,j≤n, then it follows

P^l_k(t)TB(t,s)U= p_l(t)

∑







Πkt_1j(t,s)u^j ...

Π_kt_nj(t,s)u^j





 = p_l(t)TB(t,s)P_kU

=TB(t,s)P^l_k(s)U for alls,t ∈ ^{J, (3.13)} becausep_l : J →^L(Rⁿ)is an invariant projector for (2.2). Since the matrix functionBdoes not depend onx∈ Ω, the operatorsAandBcommute and consequently the product representa- tionT(t,s) =T_A(t,s)TB(t,s)holds for alls ≤t. We arrive at

T(t,s)P^l_i(s)U ^(3.9)= S _{Z t}

s a

TB(t,s)P^l_i(s)U^(3.13)= S _{Z t}

s a

P^l_i(t)TB(t,s)U

(3.12)

= S _{Z t}

s a

P_ip_l(t)TB(t,s)U=P_iS _{Z t}

s a

p_l(t)TB(t,s)U due to Lemma3.4. This allows us to continue

T(t,s)P^l_i(s)U ^(3.9)= P_i

∑

k∈^N

e^λkRstaP_kp_l(t)TB(t,s)U

(3.12)

= P_i

∑

k∈^N

e^λkRstap_l(t)P_kTB(t,s)U

= P_ip_l(t)

∑

k∈^N

e^λkRstaP_kTB(t,s)U

(3.12)

= P^l_i(t)

∑

k∈^N

e^λkRstaP_kTB(t,s)U

(3.9)

= P^l_i(t)TA(t,s)TB(t,s)U =P_i^l(t)T(t,s)U for all s≤^t.

Consequently, X_i^{l :}= (t,U)∈ ^J×X: U∈ ^R(P^l_i(t)) are finite-dimensional vector bundles being invariant w.r.t. (3.11). On every Whitney sum X_k¹⊕ · · · ⊕ X_k^m ⊆ ^J×Xthe dynamics is determined by the linear ODE

˙

u = [λka(t) +B(t)]u for allk∈^N

inRⁿwith evolution operator T^k(t,s):= T(t,s)P_k. It consequently follows that T(t,s) =

∑

k∈^N

T^k(t,s) for all s≤^t and thusΣJ(_aA+_B) =^S_k∈NΣJ(aλ_k+B).

Corollary 3.9. If a(t)≡^α>_{0on J, then}

ΣJ(aA+B) =α ^[

k∈^N

{^λk}+ΣJ(B).

Proof. For such constant functionsathe dichotomy spectrum of (3.11) becomes ΣJ(αA+B) = ^[

k∈^N

ΣJ(αλ_k+B)^(2.1)= ^[

k∈^N

ΣJ(αλ_k) +ΣJ(B)

and this implies the claim.

4 Exponential decay

Our spectral theory obtained above provides examples well-suited to illustrate a nonau- tonomous version of [1, Lemma 5]. This vital result ensures that forward solutions to time- variant parabolic evolutionary equations cannot decay to 0 faster than exponentially.

We actually consider abstract semi-linear evolutionary equations

˙

u=A(t)u+F(t,u) (E)

in a Banach spaceX. Here,t∈ ^J is from an interval J ⊆^Runbounded above. Let us suppose that the linear part (L) induces an evolution familyT(t,s)∈ ^L(X),s≤t, with the properties:

(L₁) ΣJ(A) =^S_k∈N[λ⁻_k ,λ⁺_k ]⊆ (−∞,α0]for someα0∈^R;

(L₂) there exist real sequences(αk)_k∈N,(βk)_k∈Nwith

· · · <_α2< _β2 <_α1< _β1 <_α0 such thatΣJ(A)⊆^Sk∈^N(β_k,α_k−1)(cf. Fig.4.1);

(L3) the invariant projectors Pk : J → ^L(X) associated to the spectral intervals [λ⁻_k ,λ⁺_k ], k ∈N, are complete.

R λ⁻₁ λ⁺₁

λ⁻₂ λ⁺₂ λ⁻_n−1 λ⁺_n−1

λ⁻_n λ⁺_n λ⁻_n+1 λ⁺_n+1

β₁ α₁ β₂

α₂ β_n−1

α_n−1 β_n

α_n α₀

Figure 4.1: Dichotomy spectrumΣJ(A)of the linear part (L) (in red) and the gap intervals[αn,βn],n∈ ^N

Concerning the continuous nonlinearityF: J×X→Xin (E) let us assume that (N) _F(t, 0)≡^{0 on J} and there exists an L≥0 such that

k^F(t,u)−F(t, ¯u)k ≤^Lk^u−^u¯k ^{for allt}∈ ^{J, u, ¯u}∈X.

The mild solution to (E) satisfying u(τ) = u0 is denoted by u(·^;τ,u0) : [τ,∞) → ^{X for an} initial timeτ∈ ^J and an initial stateu₀ ∈X.

Let the center of the gap intervals [α_k,β_k] (cf. Fig.4.1) be denoted byγ_k := ^αk+₂^βk and we introduce thepseudo-stable fiber bundles

Wk :=

(τ,u₀)∈ ^J×^{X: lim}_t

→∞e^γk(τ−t)k^u(t;τ,u₀)k=0

for all k∈^N.

These sets clearly satisfy the inclusionsWk+1 ⊆ Wk for allk ∈^N.

Notice that a mild solutionνto (E) is said to besmall, if for everyγ∈^{Rone has}

tlim→∞e^γtk^ν(t)k=0.

While small solutions can occur e.g. in the context of delay differential equations (we refer to [6, pp. 74ff., Section 3.3]), the next result rules out nontrivial small solutions in our setting.

Theorem 4.1. Under the above assumptions(L)and(N)with 0≤^L < inf

k∈^N

β_k−α_k

6K_k (4.1)

one has ^T_k∈NWk = J× {⁰}, i.e. for every nontrivial (mild) solution ν : [τ,∞) → ^{Xto (E) there} exists a k∈^{Nsuch that}(τ,ν(τ))∈ Wk+1\ Wk.

In few words, Theorem 4.1 implies that for every nontrivial mild solution ν there exists a γ ∈ ^R with lim sup_t→∞e^γ(τ−t)k^ν(t)k > 0. This means that (E) cannot have nontrivial solutions decaying to 0 faster than exponentially. As the subsequent proof demonstrates, our Theorem4.1is essentially a corollary of the classical Hadamard–Perron theorem on stable manifolds. Concerning a version appropriate for our purposes we refer to [13, Theorem 2.4(a)].

Proof. Letτ∈ ^J be arbitrary. Given γ∈^Rit is easy to see that the sets Bτ,γ :=

φ∈ ^C[τ,∞;X): lim

t→∞e^γ(τ−t)k^φ(t)k=0

with the normk^φk_τ,γ ^:=sup_τ≤te^γ(τ−t)k^φ(t)kare Banach spaces.

(I) Our assumptions (L1)–(L2) on the dichotomy spectrum ensure that for every k ∈ ^N there exist realsK_k ≥1 and an invariant projectorP⁺_k : J →^L(X)so that the estimates

T(t,s)P⁺_k (s) ≤^Kke^αk(t−s),

T¯(s,t)P⁻_k(t)≤^Kke^βk(s−t) fors ≤^{t (4.2)} are fulfilled with the complementary projector P⁻_k (t) := id_X−P⁺_k (t). For every particular growth rateγ:= ^αk+₂^βk ∈ (α_k,β_k),k∈ ^Nfixed, let us define the operators

S_τ ∈ ^L(X,B_τ,γ), S_τu₀ :=T(·^,τ)P⁺_k(τ)u₀, Tτ :B_τ,γ →^Bτ,γ, Tτ(φ):=

Z _·

τ T(·^,s)P⁺_k (s)F(s,φ(s))ds

−

Z _∞

·

T¯(·^,s)_P⁻_k(s)_F(s,φ(s))ds.

They are well-studied in the literature (e.g. [1,11,13,15]) whenB_τ,γ is replaced by the space of all continuous functionsφsatisfyingk^φk_τ,γ <∞. Thus, it remains to show that the mappings Sτ,Tτ are well-defined.

First, for everyu0∈^Xone has the limit relation k(Sτu₀)(t)k^eγ(τ−t) =T(t,τ)P⁺_k (τ)u₀

e^{γ(τ−t)(4.2)}≤ ^Kke^{(αk−γ)(t−τ)}k^u0k −−→_t

→∞ 0 and thereforeSτu₀∈ ^Bτ,γ.

Second, concerning the operator Tτ choose an arbitrary φ ∈ ^Bτ,γ. This ensures that for everyε>0 there exists aT ≥^{τsuch that}

max

K_kL

γ−^αk, K_kL β_k−^γ

e^γ(τ−t)k^φ(t)k< ^ε

3 for allt ≥^{T. (4.3)}

Because of(N)we arrive at the estimate k^Tτ(φ)(t)k^(4.2)≤ ^KkL^{Z t}

τ e^αk(t−s)k^φ(s)k^ds+K_kL^{Z ∞}

t e^βk(t−s)k^φ(s)k^ds

≤ ^KkL^{Z T}

τe^αk(t−s)e^γ(s−τ)dsk^φk_τ,γ+K_kL^{Z t}

T e^αk(t−s)k^φ(s)k^ds +K_kL^{Z ∞}

t e^βk(t−s)k^φ(s)k^{ds for all τ}≤^t.

This, in turn, implies

k^Tτ(φ)(t)k^eγ(τ−t) ≤ _γ^KkL

−^αk

he^{(αk−γ)(t−T)}−^{e(αk−γ)(t−τ)i}k^φk_τ,γ

+K_kL^{Z t}

T e^αk(t−s)k^φ(s)k^{eγ(τ−s)eγ(s−t)ds} +K_kL^{Z ∞}

t e^βk(t−s)k^φ(s)k^{eγ(τ−s)eγ(s−t)ds}

(4.3)

< ^KkL

γ−^αke^{(αk−γ)(t−T)}k^φkτ,γ+^γ−αk

3 ε Z _t

T e^αk(t−s)e^γ(s−t)ds + ^βk−^γ

3 ε Z _∞

t e^βk(t−s)e^γ(s−t)ds

< ^KkL

γ−^αke^{(αk−γ)(t−T)}k^φkτ,γ+ ^ε 3+ ^ε

3 for all T≤^t

and due toα_k <_γ there is aT⁰ ≥ ^{Tsuch that} _γ^K₋^kL_αk^{e(αk−γ)(t−T)}k^φk_τ,γ < ₃^ε holds for allt ≥^T0. Consequently,k^Tτ(φ)(t)k^eγ(τ−t) <_{εfor everyt} ≥^{T0, i.e.T}τ(φ)∈ ^Bτ,γ.

(II) Thanks to (I) the Lyapunov–Perron operator

Lτ :B_τ,γ×^X→ ^Bτ,γ, Lτ(φ,u₀):= Sτu₀+Tτ(φ)

is well-defined. As in the proof of [13, Theorem 2.4] one establishes that (4.1) guarantees Lτ

to be a uniform contraction in the first argument. From the contraction mapping theorem we deduce a unique fixed-point φ^∗_τ(u₀) ∈ ^Bτ,γ. Setting w_k(τ,u₀) := P⁻_k (τ) φ^∗_τ(u₀)(τ) one obtains a functionw_k : J×X→Xfulfillingw_k(τ, 0)≡^{0 on J} and a global Lipschitz condition with constant<1. Moreover, it holds the representation

Wk =(τ,ξ+w_k(τ,ξ))∈ ^J×^{X: ξ} ∈^R(P⁺_k (τ)) .

(III) After these preparations the actual proof is quite immediate. Indeed, let us suppose that ν:[τ,∞)→Xis a mild solution of (E) which is contained in all Wk,k ∈N. This implies ν(τ) =P⁺_k (τ)ν(τ) +w_k(τ,P⁺_k (τ)ν(τ))and consequently

k^ν(τ)k ≤P⁺_k (τ)ν(τ)+w_k(τ,P⁺_k(τ)ν(τ))−^wk(τ, 0)≤²P⁺_k (τ)ν(τ)−−→_k

→∞ 0,