PII. S0161171203209157 http://ijmms.hindawi.com © Hindawi Publishing Corp.
LINEAR NEUTRAL PARTIAL DIFFERENTIAL EQUATIONS:
A SEMIGROUP APPROACH
RAINER NAGEL and NGUYEN THIEU HUY
Received 17 September 2002
We study linear neutral PDEs of the form(∂/∂t)F ut=BF ut+Φut,t≥0;u0(t)= ϕ(t),t≤0, where the function u(·) takes values in a Banach spaceX. Under appropriate conditions on the difference operatorFand the delay operatorΦ, we construct aC0-semigroup onC0(R−, X)yielding the solutions of the equation. 2000 Mathematics Subject Classification: 34K60, 47D06, 34K30, 34G10.
1. Introduction. In this paper, we study linear neutral PDEs of the form
∂
∂tF ut=BF ut+Φut fort≥0, u0(t)=ϕ(t) fort≤0.
(1.1)
Here,Bis some linear partial differential operator, while the operatorsF and
Φare calleddifference operatoranddelay operator, respectively. We refer to Hale [5,6], Wu [12, Chapter 2.3], Wu and Xia [13], and Adimy and Ezzinbi [1] for concrete examples and propose an abstract treatment of these equations. For that purpose, we choose a Banach spaceXand consider the solutionu(·)as a function fromRtoX. Then, the correspondinghistory functionis defined as
ut(s):=u(t+s) ∀t≥0, s≤0. (1.2)
Moreover,Bis a linear operator onX(representing the partial differential oper-ator), whileFandΦare linear operators from anX-valued function space; for example,C0(R−, X)intoX. More precisely, we make the following assumption
throughout the paper.
Assumption1.1. On the Banach spacesXandE:=C0(R−, X), we consider
the following operators:
(i) let (B, D(B)) be the generator of a strongly continuous semigroup (etB)t
≥0onXsatisfyingetB ≤Meω1t for some constantsM≥1 and
ω1∈R;
Under these assumptions, we will solve (1.1) by constructing an appropriate strongly continuous semigroup on the spaceE. This semigroup will be obtained by proving that a certain operator (seeDefinition 2.3) satisfies the Hille-Yosida conditions as long as we can write the difference operator asF=δ0−Ψ with
Ψbeing “small” (see (2.7)). If the delay and difference operators act only on a finite interval[−r ,0], it can be shown that the smallness ofΨcan be replaced by the condition “having no mass in 0” (seeDefinition 3.1).
In the case of ordinary neutral functional differential equations on finite-dimensional spaces X, we refer the readers to Hale and Verduyn Lunel [7, Chapter 9], Engel [3], and Kappel and Zhang [9, 10] for results about well-posedness and asymptotic behavior of the solutions as well as the use of the condition “having no mass in 0” (or, “nonatomic at zero,” seeRemark 3.2). In the case of infinite-dimensional spacesX, such a condition appeared in [11] (see also Datko [2]), where the generator property has been shown under dissi-pativity conditions for ordinary neutral functional differential equations. Hale [5,6] and Wu [12, Chapter 2.3] assumedBto generate an analytic semigroup and also obtained a semigroup solving (1.1) in a mild sense ifΨis nonatomic at zero.
2. Neutral semigroups with infinite delay. Under the assumptions from Section 1, we consider the Banach spaceE:=C0(R−, X)endowed with the
sup-norm and the operator(Gm, D(Gm))onE, defined by
DGm:=f∈E∩C1R
−, X:f∈E,
Gmf:=f forf∈DGm. (2.1)
We are now looking for various restrictions of this (maximal) operator yield-ing generators of strongly continuous semigroups. We start with a simple case.
Definition2.1. On the spaceE=C0(R−, X), we define the operatorTB,0(t)
by
TB,0(t)f
(s)=
f (s+t), s+t≤0,
e(t+s)Bf (0), s+t≥0 (2.2)
forf∈Eandt≥0.
Moreover, we define the operator(GB,0, D(GB,0))by
DGB,0
:=f∈DGm:f (0)∈D(B), f∈E, f(0)=Bf (0),
GB,0f:=f forf∈D
GB,0
. (2.3)
We then have the following properties ofGB,0and(TB,0(t))t≥0which can be
Proposition2.2. The following assertions hold:
(i) (TB,0(t))t≥0is a strongly continuous semigroup on the spaceEwith the generator(GB,0, D(GB,0));
(ii) the set{λ∈C: Reλ >0, λ∈ρ(B)}is contained in ρ(GB,0). Moreover, forλin this set, the resolvent is given by
Rλ, GB,0f(t)=eλtR(λ, B)f (0)+ 0
t e
λ(t−ξ)f (ξ)dξ forf∈E, t≤0;
(2.4)
(iii) the semigroup(TB,0(t))t≥0satisfies
TB,0(t)≤Meω2t, t≥0 (2.5)
with ω2 := max{0, ω1} for the constants M and ω1 appearing in Assumption 1.1.
We now take the delay operator Φ and the difference operator F (see Assumption 1.1) to define a different restriction of the operatorGm.
Definition2.3. The operatorGB,F ,Φis defined by
GB,F ,Φf:=fon the domain
DGB,F ,Φ:=f∈DGm:F f∈D(B), Ff=BF f+Φf. (2.6)
Our aim is to find conditions onFsuch that the operatorGB,F ,Φbecomes the generator of a strongly continuous semigroup. To do so, we writeFin the form
F f:=f (0)−Ψf , f∈E, (2.7)
for some bounded linear operatorΨ:E→X. The domain ofGB,F ,Φcan then be rewritten as
DGB,F ,Φ
=f∈DGm:f (0)−Ψf∈D(B), f(0)=Bf (0)−Ψf+Φf+Ψf. (2.8)
It is now our main result that if the operatorΨ is “small,” thenGB,F ,Φ is densely defined and satisfies the Hille-Yosida estimates; hence, generates a strongly continuous semigroup.
For that purpose and forλ∈Csatisfying Reλ >0, we define the operator eλ:X→Eby
Theorem 2.4. Assume that the difference operator F is of the form (2.7) such thatΨsatisfies the conditionΨ<1. Then, the following assertions hold: (i) λ∈ρ(GB,F ,Φ)for eachλ > ω2+MΦ/(1−Ψ)(with the constantsω2 andMas inProposition 2.2). For suchλ, the resolvent ofGB,F ,Φhas the form
Rλ, GB,F ,Φf=eλΨRλ, GB,F ,Φ+R(λ, B)ΦRλ, GB,F ,Φ−Ψf +Rλ, GB,0f forf∈E;
(2.10)
(ii) forL:=(M+MΨ)/(1−Ψ)andλ0:=ω2+MΦ/(1−Ψ),
R
λ, GB,F ,Φ≤λ−Lλ 0
forλ > λ0; (2.11)
(iii) forλ > ω0:=max{2λ0, ω2+LΦ}andP:=3e[(M+L)Ψ +2M+1], it follows that
Rλ, GB,F ,Φn≤ P λ−ω0
n ∀n∈N; (2.12)
(iv) the operatorGB,F ,Φis densely defined.
Proof. (i) We first observe that foru, f ∈Eandλ∈C, we have thatu∈
D(Gm)andλu−Gmu=f if and only ifuandfsatisfy
u(t)=eλ(t−s)u(s)+ s
te
λ(t−ξ)f (ξ)dξ fort≤s≤0. (2.13)
Note that forλ > ω2and by (2.4),
u(t)=eλtΨu+R(λ, B)f (0)+Φu−Ψf
+ 0
t
eλ(t−ξ)f (ξ)dξ fort≤0 (2.14)
is equivalent to
u=eλΨu+R(λ, B)Φu−eλR(λ, B)Ψf+Rλ, GB,0
f . (2.15)
If, for eachf∈Eandλ > λ0, the equation (2.15) has a unique solutionu∈E,
thenu(0)=Ψu+R(λ, B)(f (0)+Φu−Ψf ). This is equivalent to
or
u(0)=Bu(0)−Ψu+Φu+Ψu. (2.17)
Hence, by the above observation and the definition of GB,F ,Φ, we have that u∈D(GB,F ,Φ)andu=R(λ, GB,Φ,F)f. Therefore, to prove (i), we have to verify that forλ > λ0and eachf∈E, (2.14) has a unique solutionu∈E.
LetMλ:E→Ebe the linear operator defined as
Mλ:=eλΨ+R(λ, B)Φ. (2.18)
Sinceλ > ω2+MΦ/(1−Ψ), we have thatMλis bounded and satisfies
Mλ≤ Ψ+λM−ωΦ 2
<1. (2.19)
Therefore, the operatorI−Mλis invertible, and (2.14) has a unique solution u=(I−Mλ)−1(R(λ, GB,0)f−eλR(λ, B)Ψf ). Thus,
Rλ, GB,F ,Φf=MλRλ, GB,F ,Φf−eλR(λ, B)Ψf+Rλ, GB,0
f (2.20)
and (2.10) follows.
(ii) By the Neumann series(I−Mλ)−1=∞n=0Mλn, we have that
R
λ, GB,F ,Φ=
∞
n=0
Mλn
Rλ, GB,0−eλR(λ, B)Ψ
≤ ∞
n=0 Mn
λ
M+MΨ
λ−ω2
≤M+MΨ λ−ω2
∞
n=0
Ψ+MΦ
λ−ω2 n
= M+MΨ
1−Ψλ−ω2−MΦ/1−Ψ
= L
λ−λ0
.
(2.21)
(iii) Forλ > λ0andu:=R(λ, GB,F ,Φ)f, we have
u(t)=eλtΨRλ, GB,F ,Φf+R(λ, B)ΦRλ, GB,F ,Φf−Ψf+f (0)
+ 0
t
We extenduandf to functions onRby
˜ u(t):=
u(t) fort≤0,
eλtg(t) fort >0,
˜ f (t):=
f (t) fort≤0,
−eλtg(t) fort >0,
(2.23)
where we takeg(t):=u(0)+t
0ϕ(τ)dτwith
ϕ(t):=
6ttλ2−λλu(0)−1
2f (0)
+[λt−1]f (0) for 0≤t≤1
λ,
0 fort≥ 1
λ.
(2.24)
Then,gis continuously differentiable with compact support contained in the interval[0,1/λ]satisfyingg(0)=u(0),g(0)= −f (0), and
eλtg(t)≤3e(M+L)Ψ+2M+1f (2.25)
forλ >max{2λ0, ω2+LΦ}and all t∈R+. Hence, the functions ˜u and ˜f
defined by (2.23) belong to the Banach spaceC0(R, X)and satisfy
˜
u(t)=eλ(t−s)u(s)˜ + s
t
eλ(t−ξ)f (ξ)dξ˜ fort≤s, (2.26) f˜≤3e(M+L)Ψ+2M+1f. (2.27)
We now look at the left translation semigroup(T (t))t˜ ≥0on the Banach space
˜
E:=C0(R, X), that is,
˜
T (t)f˜(s):=f (s˜ +t) ∀f˜∈E, s˜ ∈R, t≥0. (2.28)
This semigroup is strongly continuous on ˜E and its generator is ˜Gm:=d/ds on the domainD(Gm)˜ := {f ∈E˜∩C1(R, X):f∈E˜}(see [4, Chapter II.2]). Furthermore, we observe that forv, w∈E˜andλ∈C, we have thatv∈D(Gm)˜ andλv−Gmv˜ =wif and only ifvandwsatisfy (2.26). Sinceλ∈ρ(Gm)˜ for λ > λ0, we obtain that ˜u=R(λ,Gm)˜ f˜forλ > λ0, where ˜uand ˜fare defined as
in (2.23).
Therefore, by (2.23), we have that
Rλ, GB,F ,Φf(t)=u(t)=u(t)˜ =Rλ,Gm˜ f˜(t) (2.29)
By induction, we obtain
Rλ, GB,F ,Φnf(t)=Rλ,Gm˜ nf˜(t) fort≤0, λ > ω0. (2.30)
Using the fact that ˜Gmis the generator of the strongly continuous semigroup (T (t))t˜ ≥0on ˜E, and by inequality (2.27), we have
Rλ, GB,F ,Φnf
(t)=Rλ,Gm˜ nf˜(t)
≤f˜
λn ≤
3e(M+L)Ψ+2M+1
λ−ω0
n f
(2.31)
for allt≤0,λ > ω0, and alln∈N. Therefore, puttingP:=3e[(M+L)Ψ +
2M+1], we obtain
Rλ, GB,F ,Φnf≤ P λ−ω0
nf forλ > ω0, n∈N. (2.32)
(iv) Forλ > λ0, we consider the operatorS:E→Edefined by
Sf:= −eλR(λ, B)Ψf+Rλ, GB,0f , f∈E. (2.33)
Observe that if its range ImS is dense in E, then we have, by (2.20), that D(GB,F ,Φ)=R(λ, GB,F ,Φ)E=(I−Mλ)−1ImSis dense inE. Therefore, it is enough to verify thatImSis dense inE. SinceD(Gm)is dense inE, we only need to show thatImS⊃D(Gm)=D(λ−Gm).
In fact, foru∈D(λ−Gm), there existsf∈Esuch that
u(t)=eλtu(0)+ 0
t e
λ(t−ξ)f (ξ)dξ fort≤0. (2.34)
Since the operatorB is densely defined, there exists a sequence(yn)⊂D(B) such that limn→∞yn=u(0). Let(xn)⊂Xbe a sequence such thatR(λ, B)xn=
yn.
For eachn∈N, we choose a real-valued, continuous functionαn(t) with support contained in [max{−1/n,−1/nxn},0] satisfying αn(0)= 1 and supt≤0|αn(t)| ≤1.
By the conditionΨ<1, we have that the functions
belong toE. Moreover, these functions satisfy
fn(t)=αn(t)xn+Ψfn−f (0)+f (t),
fn(0)−Ψfn=xn, fn≤xn1−+Ψ2f.
(2.36)
We now put
un(t):=eλtR(λ, B)fn(0)−Ψfn+ 0
t e
λ(t−ξ)fn(ξ)dξ. (2.37)
Then,un=Sfn; henceun∈ImS, and forλ > λ0, we obtain
un(t)−u(t)
=eλtR(λ, B)xn−u(0)+ 0
t e
λ(t−ξ)fn(ξ)−f (ξ)dξ
≤yn−u(0)+ 0
t
αn(t)xn+Ψfn+f (0)dt
≤yn−u(0)+ 0
max{−1/n,−1/nxn}
xn+Ψxn+2f
1−Ψ +f
dt
≤yn−u(0)+n 1 1−Ψ+
1+Ψf
n1−Ψ
n→∞
→0 uniformly∀t∈R−.
(2.38)
This means that limn→∞un=u. Thus,ImSis dense inE.
The Hille-Yosida theorem now yields the following corollary.
Corollary2.5. Let the difference operatorF have the form (2.7) withΨ satisfyingΨ<1. Then the operatorGB,F ,Φ generates a strongly continuous semigroup(TB,F ,Φ(t))t≥0onEsatisfying
TB,F ,Φ(t)≤P eω0t, t≥0, (2.39)
where the constantsP andω0are defined as inTheorem 2.4.
We now conclude this section by a corollary about well-posedness of (1.1). To this end, we denote bytut(·, ϕ) the classical solution of (1.1) corre-sponding to the initial conditionu0=ϕ, that is,tut(·, ϕ)is continuously
Corollary2.6. Assume that the difference operatorF is of the form (2.7) such thatΨsatisfiesΨ<1. Then, (1.1) is well posed. More precisely, for every ϕ∈D(GB,F ,Φ), there exists a unique classical solutionut(·, ϕ)of (1.1) given by
ut(·, ϕ)=TB,F ,Φ(t)ϕ, (2.40)
and for every sequence(ϕn)n∈N⊂D(GB,F ,Φ)satisfyinglimn→∞ϕn=0,
lim n→∞ut
·, ϕn=0 (2.41)
uniformly in compact intervals.
Proof. ByCorollary 2.5, the operator(GB,F ,Φ, D(GB,F ,Φ))defined by (2.6) is
the generator of the strongly continuous semigroup(TB,F ,Φ(t))t≥0.
For ϕ ∈ D(GB,F ,Φ), we put ut := TB,F ,Φ(t)ϕ. Then, it is clear that ut ∈ D(GB,F ,Φ)⊂D(Gm). We now show thatutsatisfies (1.1). Indeed, we have
d
dtF ut=limh→0
F ut+h−F ut h =hlim→0
F T (t+h)ϕ−F T (t)ϕ h
=Flim h→0
T (h)T (t)ϕ−T (t)ϕ
h =F GB,F ,ΦT (t)ϕ
=BF T (t)ϕ+ΦT (t)ϕ=BF ut+Φut.
(2.42)
For the uniqueness of the solution, we prove that ifvtis a classical solution of (1.1) satisfyingv0=0, thenvt=0 for allt≥0. In fact, sincevtsatisfies (1.1)
andvt∈D(Gm), we have that
BF vt+Φvt= d
dtF vt=hlim→0
F vt+h−F vt h =Flimh→0
vt+h−vt h =F v
t. (2.43)
Therefore,vt∈D(GB,F ,Φ)satisfies the Cauchy problem
d
dtvt=GB,F ,Φvt fort≥0, v0=0.
(2.44)
SinceGB,F ,Φis the generator of a strongly continuous semigroup, this Cauchy problem has a unique solutionvt=0 (see [4, Theorem II.6.7]).
Finally, the last assertion, called thecontinuous dependence on the initial dataof the solutions, follows from the uniform boundedness of the strongly continuous semigroup(TB,F ,Φ(t))t≥0on compact intervals.
3. Neutral semigroups with finite delay. In this section, we study (1.1) on a finite delay interval[−r ,0], that is,
∂
∂tF ut=BF ut+Φut fort≥0, u0=ϕ∈C
[−r ,0], X.
We again assume the difference operatorF to be given as in (2.7), that is,
F ϕ=ϕ(0)−Ψϕ, ϕ∈C[−r ,0], X (3.2)
for some bounded linear operatorΨ:C([−r ,0], X)→X. However, instead of assumingΨto be “small,” we suppose thatΨhas no mass in0 (seeDefinition 3.1). Our main idea is to renorm the space C([−r ,0], X)such that, with the new equivalent norm, the norm ofΨ is small, so we can adapt the arguments from the previous section. This idea has been used, for example, by Schwarz [11] to study ordinary neutral functional differential equations via dissipativity conditions. We begin with the definition of “no mass in 0.”
Definition3.1. A bounded linear operatorΨ∈ᏸ(C([−r ,0], X), X)is said
to haveno mass in 0 if for every >0, there exists a positive numberδ≤r such that
Ψ(f )X≤f∞ ∀f∈C[−r ,0], Xsatisfying suppf⊆[−δ,0]. (3.3)
Remark3.2. This definition is taken from [11, Definition II.2.1]. We note
that, ifΨ∈ᏸ(C([−r ,0], X), X)has the form
Ψ(f )= 0
−r
dη(θ)f (θ) (3.4)
for some functionη(·)of bounded variation, then the above definition is equiv-alent to the fact that the functionη(·)is nonatomic at 0 in the sense of Hale and Verduyn Lunel [7, Chapter 9.2] or Wu [12, Chapter 2.3].
We are now prepared to renorm the spaceC :=C([−r ,0], X). Indeed, for each positive numberω, the new norm·ωdefined by
fω:= sup
−r≤s≤0
f (s)e−ωs
X, f∈C (3.5)
is equivalent to the supnorm. Furthermore,Cωdenotes the spaceC([−r ,0], X) endowed with the norm·ω.
Lemma3.3. Let the operatorΨ ∈ᏸ(C, X)have no mass in 0. Then there exists a positive numberωsuch that the norm of the operatorΨ, as a bounded linear operator fromCωintoX, is smaller than 1.
Proof. We first prove that there exists a numberω >0 such that
Ψ(f )X≤12 ∀f∈C[−r ,0], Xsatisfyingf (s)≤eωs ∀s∈[−r ,0]. (3.6)
Indeed, sinceΨhas no mass in 0, there exists a positive numberδ≤rsuch that
Ψ< (1/4)eδω. Now, for a givenf∈C([−r ,0], X)satisfyingf (s) X≤eωs for alls∈[−r ,0], we prove thatΨ(f )X≤1/2. For that purpose, we put
f1(s):=
f (s) fors∈[−r ,−δ],
f (−δ) otherwise, (3.7)
andf2(s):=f (s)−f1(s). Then, suppf2⊆[−δ,0]. Therefore, we have that Ψ(f )X≤Ψf1X+Ψ
f2X
≤ Ψf1∞+1
8f1∞+f∞
<1 2
(3.8)
and (3.6) follows.
Denote byΨω the norm ofΨas a bounded linear operator inᏸ(Cω, X). Then, by inequality (3.6), we have that
Ψω= sup
fω≤1
Ψf = sup
sup
−r≤s≤0
f (s)e−ωs≤1Ψ
f ≤12<1. (3.9)
This renorming allows us to adapt the arguments from the proof ofTheorem 2.4to prove that the operator corresponding to (3.1) is the generator of aC0
-semigroup on the spaceC([−r ,0], X). Note that the generator property of an operator is preserved when passing to an equivalent norm.
Theorem 3.4. Assume that the difference operatorF has the formF ϕ=
ϕ(0)−Ψϕwith the bounded linear operatorΨ:C([−r ,0], X)→Xhaving no mass in0, and let the operatorBgenerate a strongly continuous semigroup on X. Then, the operator(G, D(G))defined by
Gf:=f on the domain
D(G):=f∈C[−r ,0], X∩C1[−r ,0], X:F f∈D(B), F f=BF f+Φf (3.10)
is the generator of a strongly continuous semigroup(T (t))t≥0onC([−r ,0], X).
Proof. LetCωbe the spaceC([−r ,0], X)normed by the new norm · ω
forωas inLemma 3.3. Then, the norm of the operatorΨ, as a bounded lin-ear operator fromCωintoX, is smaller than 1. Therefore, as inTheorem 2.4, we show that the operator(G, D(G))defined by (3.10) is densely defined and satisfies the Hille-Yosida estimates; hence, it generates a strongly continuous semigroup.
Corollary3.5. Assume that the difference operatorF is of the form (3.2) such thatΨhas no mass in0. Then, for everyϕ∈D(G), there exists a unique classical solutionut(·, ϕ)of (3.1), given byut(·, ϕ)=T (t)ϕ, where the strongly continuous semigroup(T (t))t≥0is generated by the operatorGas inTheorem 3.4. Moreover, for every sequence(ϕn)n∈N⊂D(G)satisfyinglimn→∞ϕn=0,
limn→∞ut(·, ϕn)=0uniformly in compact intervals.
Having established the well-posedness of (3.1), we now consider the robust-ness of the exponential stability of the solution semigroup. This can be done by using the constants appearing in the Hille-Yosida estimates of the operatorG.
Corollary3.6. Let the assumptions ofTheorem 3.4be satisfied. In addition, let the operatorBgenerate an exponentially stableC0-semigroup and the norm of the operatorΨsatisfyΨ<1. Then, if the norm of the delay operatorΦis sufficiently small, the solution semigroup(T (t))t≥0generated by(G, D(G))is exponentially stable.
Proof. We note that in the case of the finite delay interval[−r ,0], the
op-eratorseλdefined as in (2.9) are well defined for allλ∈C, and the exponentω2
in the exponential estimate (2.5) can be chosen asω2:=ω1with the constant
M being replaced byK:=max{M, Me−ω1r}, where the constantω1appears
inAssumption 1.1. Therefore, for (3.1) on the finite delay interval[−r ,0], we can adapt the arguments in the proof ofTheorem 2.4to obtain an analogue ofCorollary 2.5, that is, the generator(G, D(G))defined by (3.10) generates a strongly continuous semigroup(T (t))t≥0satisfying
T (t)≤P eω0t, t≥0 (3.11)
with the constantPdefined as inTheorem 2.4and the constant
ω0:=max
2ω1+
2KΦ
1−Ψ, ω1+
K+KΨ
1−Ψ Φ
. (3.12)
Now, the assumption that(etB)t
≥0is exponentially stable means thatω1<0.
Therefore, ifΦ<−ω1(1−Ψ)/(K+KΨ), then the solution semigroup is
exponentially stable.
Remark3.7. In order to show the robustness of the exponential stability
of the solution semigroup as in the above corollary, we need the condition
Acknowledgment. The second author gratefully acknowledges the
sup-port of the Vietnamese government.
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E-mail address:[email protected]
Nguyen Thieu Huy: Arbeitsgemeinschaft Funktionalanalysis (AGFA) Mathematisches Institut der Universität Tübingen, Auf der morgenstelle 10, 72076 Tübingen, Germany
