Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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598 W.M. Ruess

4. As Theorem 2.4 touches upon exponential asymptotic stability of the solution

semigroup to (PFDE)

lin

, we note that there are instances where this can be read

directly from the relevant parameters of

B and

F, cf. [25, Prop. 4.4].

Examples 1. Given a bounded open domain Ω ⊂ R

, 1 ≤ N ≤ 3, of class C

we let X := C

(Ω) := {u ∈ C(

Ω) | u

|∂Ω

=0} with sup-norm, and consider the

Dirichlet-Laplacian in X,



D(Δ

)={u ∈ C

(Ω) | Δu ∈ C

(Ω)}

u := Δu ∈D(Ω)



(2.3)

and note that, given any d>0, (−dΔ

)ism−ω

-accretive for some ω

< 0inX

(cf. [2, Section 6.1]).

2. Given real numbers α ≤ 0,β ≥ 0, [α, β] denotes the order interval of all u ∈ X

such that α ≤ u(ω) ≤ β for all ω ∈

Ω.

For a continuous and monotonically non-decreasing function β : R → R ,

with β(0) = 0, we denote by

β : X → X its realization in X = C

(Ω), given

by (

βu)(ω)=β(u(ω)),ω∈

Ω, and consider the following nonlinearly perturbed

model of a diﬀusive population with delay in the birth process:

⎧

⎪

⎨

⎪

⎩

∂

∂t

u(x, t)+(−dΔ

β)u(x, t)

= au(x, t)

1 −bu(x, t) −



−1

u(x, t + r(s))dη(s)

|[−R,0]

= ϕ ∈ C([−R, 0]; C

(Ω))

(2.4)

where a, b, d > 0,η is a positive bounded regular Borel measure on [−1, 0] with

(η > 0 and) b +η =1, and r :[−1, 0] → [−R, 0] is a continuous delay function.

(For the special case of β ≡ 0, this equation serves as a model for the density of

red blood cells in an animal, cf. [10, 19, 32].)

Proposition 2.6.

(a) For al l ϕ ∈ E := C([−R, 0]; C

(Ω)) with ϕ(s) ≥ 0 for all s ∈ [−R, 0], there

exists a unique global mild solution u

to (2.4) with

(t) ∈ [0, max{1/b, ϕ(0)}]

for all t ≥ 0.

(b) If β : R → R is diﬀerentiable at 0 ∈ R , with β



(0) = ρ, and a<−ω

+ ρ,

then the zero equilibrium of (2.4) is locally exponentially stable.

(Note that, by the deﬁnition of the function β, ρ ≥ 0.)

Remarks 2.7. 1. The assertions of Proposition 2.6 actually hold for −dΔ

being

replaced by any m−ω-accretive linear operator D ⊂ C

(Ω) × C

(Ω), and the

operator

β being replaced by any (single-valued) m-accretive operator C : D(C) ⊂

X → X with D(C) ⊃ D(D),C(0) = 0, and such that B := D + C is m−α-

accretive, some α ∈ R , and such that

(i) the resolvents of B = D + C are order-preserving, and there exists

Linearized Stability 599

(ii) a D(C)-Fr´echet derivative

C ∈ B(X)ofC at 0 ∈ X : for every >0

there exists δ>0 such that x ∈ D(D), x <δ⇒

Cx −

<x,

(and, clearly, for proposition (b), such that

B = D +

C is ˜ω-accretive with ˜ω<0,

and such that a<−˜ω).

2. With the operators B =(−dΔ

β), or, more generally, of the form B = D + C

as speciﬁed just above, assertions analogous to those of Proposition 2.6 hold as

well for the following variant of model (2.4)

⎧

⎪

⎨

⎪

⎩

˙u(t)+Bu(t)  u(t)

1+au(t) −b(u(t))

−(1 + a − b)



−R

f(s)u(t + s)ds

,t≥ 0

|[−R,0]

= ϕ,

(2.5)

as well as a variant of (2.4) with temporal averages over the past being replaced

by spatio-temporal averages, such as

⎧

⎪

⎨

⎪

⎩

˙u(t)+Bu(t)  au(t)

1 −bu(t)

−



−R



g(·−y, t − s)u(s)(y)dyds

,t≥ 0

|(−R,0]

= ϕ

(2.6)

For a discussion of the models (2.4)–(2.6) for the state space C(

Ω), and for B =

−Δ, with either Dirichlet or Neumann boundary conditions, a (partial) list of

references is [8, 10, 17, 19, 20, 32]. In the L

(Ω)-context (with Ω not necessarily

bounded), and the operator B being any m-completely accretive operator in L

(Ω),

existence and ﬂow-invariance of solutions to models (2.4)–(2.6) have been treated

in [25, Section 5].

Proof of Proposition 2.6 and Remark 2.7. As for proposition (a), note ﬁrst that

the operator B := (−dΔ

β) is m-accretive in X (as it is an additive perturbation

of an m-accretive operator by a single-valued continuous and s-accretive operator,

[3, Prop. 2.23, Thm. 16.4]). Moreover, the resolvents J

are order-preserving, so

that J

[0,α] ⊂ [0,α] for any α>0. The proof of (a) now follows from [25,

Thm. 2.5] along the lines of the corresponding proof of [25, Prop. 5.1 (a)] (with

appropriate changes due to the change from the L

- to the sup-norm).

As for proposition (b), we note (without proof here) the following general fact:

Assume that, given a Banach space X,

(i) D ⊂ X × X is m−ω-accretive linear,

(ii) C : D(C) ⊂ X → X, with D(C) ⊃ D(D) is such that B := D + C is

α-accretive, some α ∈ R , with R(I + λB) ⊃ D(B)=D(D), for λ>0 small

enough,

(iii) x

∈ D(B), and there exists

(iv)

C ∈ B(X) such that for every >0 there exists δ>0 such that x ∈ D(D),

x −x

 <δimplies

Cx −Cx

−

C(x − x

)

<x − x

,

600 W.M. Ruess

then

B := D +

C is a resolvent-diﬀerential of B at x

(in the sense of Deﬁnition

2.2).

In particular, the operator (−dΔ

+ ρI) is a resolvent-diﬀerential of the op-

erator B := (−dΔ

β)at0∈ C

(Ω).

Next, note that, (for any α>0, and) for F :

0,α

= {ϕ ∈ C([−R, 0]; C

(Ω)) |

ϕ(0) ∈ [0,α]}→C

(Ω),Fϕ= aϕ(0)[1 −bϕ(0) −



−1

ϕ(r(s))dη(s)],

Fϕ = aϕ(0) is

a D(A)-Fr´echet-derivative of F at ϕ

≡ 0(withA the solution operator of (2.4)

as in 2.C above).

Proposition (b) follows by combining the foregoing two facts with Theorem

2.4, the assumption a<−ω

+ ρ, and the fact that, for ﬁnite delay, the solu-

tion semigroup to (PFDE) is exponentially stable if, in the setting of 2.B above,

(ω + M

) < 0 ([25, Prop. 4.4]). 

3. Proof of Theorem 2.4

Step 1: Given 0 <γ

< ˜γ as in Theorem 2.4, let 

=˜γ − γ

. According to

assumptions (L) (2) and (3), given  =



, there exist δ>0, and λ

> 0,λ

ω<1,

such that

ϕ ∈

E, ϕ − ϕ

 < 4δ =⇒

Fϕ− Fϕ

−

F (ϕ − ϕ

)

<ϕ − ϕ

, (3.1)

and

0 <λ<λ

,x∈ R(I + λB), x − x

 < 4δ =⇒ (3.2)

x − J

−J

(x −x

)

≤ λ x − x

 + λη(λ; x).

(For (3.1) to hold for ϕ ∈

E, we have used assumption (L) (3), in conjunction with

the fact that, by [25, proof of Thm. 2.1, Step 3], cl D(A)=

Let ϕ ∈

E such that ϕ −ϕ

 <δ.

Choose T

> 0 such that e

γT

< 2, and let M

:= sup{y|y ∈ F (S[0,T

+1]ϕ)}

(with (S(t))

t≥0

the solution semigroup associated to A as in (S2) above).

Let η

:= min{λ

, (3(

+ |˜ω| )+1)

−1

,δ(max{(3 + M

), (Fϕ

+1)})

−1

and choose a strictly decreasing nullsequence (η

)

of positive reals η

<η

As, according to (S2), there exists a unique global mild solution to (CP)(A; ϕ;0),

there exist for T := T

+1, and all n ∈ N,η

-discrete-scheme approximations

(0 = t

,...,t

; ϕ = ϕ

,ϕ

,...,ϕ

; ψ

,...,ψ

)to(CP)(A; ϕ;0), con-

sisting of an η

-partition of the interval [0,T],

0=t

< ···<t

≤ T, t

−t

i−1

<η

,i∈{1,...,

},T−t

<η

, (3.3)

Linearized Stability 601

and elements {ϕ

= ϕ, ϕ

,...,ϕ

}, and {ψ

,...,ψ

} in E such that

∈ D(A),i ∈{1,...,

}, (3.4)

− ϕ

i−1

− t

i−1

+ Aϕ

= ψ

,i∈{1,...,

− t

i−1

) ψ

 <η

, and

such that, if the step function Φ

:[0,t

] → E is deﬁned by

(t)=



ϕt=0,

t ∈ (t

i−1

],i∈{1,...,

}, then

Φ

(t) − S(t)ϕ <η

uniformly over t ∈ [0,t

]. (3.5)

Furthermore, with this choice, we have

lim

n→∞

F (Φ

(t)) − F (S(t)ϕ) = 0 uniformly over t ∈

0,T

. (3.6)

(For all this, compare Step 3 of the proof of [25, Thm. 2.1].)

Step 2: Given n ∈ N, let N

∈ N,N

≤

, such that t

−1

≤ T

≤ t

, and let

:= t

−t

i−1

,i∈{1,...,N

}, and choose n

∈ N such that e

γ(T

+η

)

< 2, and,

according to (3.6),

F (Φ

)) − F (S(t

)ϕ) < 1 for all n ≥ n

, and all i ∈{1,...,N

}. (3.7)

From now on, we restrict ourselves to n ≥ n

, and, for the time being, suppress

the upper index n for the above discrete-scheme approximations.

Our next goal is an estimate on ϕ

− ϕ

,i∈{1,...,N

} :

With, for λ>0,J

=(I + λA)

−1

, and

=(I + λ

−1

denoting the resolvents of

A and, respectively, of

A, we ﬁrst note that, from (3.4), ϕ

= J

(ϕ

i−1

+ h

),i∈

{1,...,N

}. Thus,

− ϕ

= J

(ϕ

i−1

+ h

) − ϕ

−

(ϕ

i−1

+ h

− ϕ

) (3.8)

(ϕ

i−1

+ h

−ϕ

),i∈{1,...,N

(Here, we use that, according to [26, Thm. 2.1] (with

X = X and

E = E),

R(I + λ

A)=E for all λ>0 small enough.)

602 W.M. Ruess

As for the ﬁrst term on the right of (3.8), given λ>0 small enough, and ψ ∈

R(I + λA), from the deﬁnitions of A and

A, we have

ψ − ϕ

−

(ψ − ϕ

)

ψ)(0) − x

− (

(ψ − ϕ

))(0)

ψ)(0) = J

(ψ(0) + λF (J

ψ))

(

(ψ − ϕ

))(0) = J

((ψ − ϕ

)(0) + λ

F (

(ψ − ϕ

))

= J

+ λF ϕ

Thus,

ψ − ϕ

−

(ψ − ϕ

)

ψ)(0) − x

−(

(ψ − ϕ

))(0)

≤

(ψ(0) + λF (J

ψ)) − J

−J

(ψ(0) − x

+ λF (J

ψ))

+ λF ϕ

) −J

− J

(λF ϕ

)

+ λ

(F (J

ψ) − Fϕ

−

F (J

ψ −ϕ

))

+ λ

F (J

ψ − ϕ

−

(ψ − ϕ

))

so that

1 − λ

1 −λ˜ω

ψ − ϕ

−

(ψ − ϕ

)

(3.9)

≤

(ψ(0) + λF (J

ψ)) − J

− J

(ψ(0) − x

+ λF (J

ψ))

(3.10)

+ λF ϕ

) − J

−J

(λF ϕ

)

(3.11)

1 − λ˜ω

F (J

ψ) − Fϕ

−

F (J

ψ − ϕ

)

(3.12)

Specializing to λ = h

, and ψ = ϕ

i−1

+ h

(recall: h

= h

,ϕ

= ϕ

, and

= ψ

, and n ≥ n

), we now estimate the terms on the right of (3.9) by means

of the diﬀerentiability assumptions (3.1) and (3.2):

The term (3.10):

i−1

(0) − x

+ h

(0) + h

F (ϕ

)

≤

i−1

−ϕ

+ h

(ψ

 + F (ϕ

))

≤Φ

i−1

) −S(t

i−1

)ϕ + S(t

i−1

)ϕ −S(t

i−1

)ϕ

 + h

ψ



+ h

(F (Φ

)) − F (S(t

)ϕ) + F (S(t

)ϕ))

≤ η

+ e

γT

ϕ − ϕ

 + η

+ h

(1 + M

) < 3δ

Linearized Stability 603

(according to (3.4)–(3.7), and the choice of η

≥ η

). Thus, from (3.2),

(ϕ

i−1

(0) + h

Fϕ

) −J

−J

(ϕ

i−1

(0) − x

+ h

(0) + h

Fϕ

)

(3.13)

≤ h

(

i−1

− ϕ

+ h

ψ

 + h

Fϕ

)+h

η(h

; ϕ

i−1

(0) + h

Fϕ

)

The term (3.11): As h

Fϕ

 <η

Fϕ

 <δ,from (3.2),

+ h

Fϕ

) − J

−J

Fϕ

)

(3.14)

≤ h

Fϕ

 + h

η(h

; x

+ h

Fϕ

). (3.15)

The term (3.12): As ϕ

−ϕ

≤Φ

) − S(t

)ϕ + S(t

)ϕ − S(t

)ϕ

 <η

γ(T

+η

)

ϕ − ϕ

 < 3δ, (3.1) reveals that

F (ϕ

) −Fϕ

−

F (ϕ

− ϕ

)

≤  ϕ

−ϕ

. (3.16)

Noting that, according to the choice of η

, 0 < (1 −h

˜ω)(1 −h

(˜ω +

))

−1

≤ 2,

and (1 − h

(˜ω +

))

−1

≤

, and using the estimates (3.9)–(3.16), we conclude

from (3.8) that, for all n ≥ n

, and all i ∈{1,...,N

ϕ

− ϕ

≤

i−1

− ϕ



1+h

˜γ

+2h



+ h

ψ





1+h

˜γ

+2h



+2h

ϕ

−ϕ

 +2h

(Fϕ

 + Fϕ

) (3.17)

+2h

(η(h

; ϕ

i−1

(0) + h

Fϕ

)+η(h

; x

+ h

Fϕ

)

With regard to this inequality, we note the following estimates:

First, as Fϕ

= F (Φ

)), we have

Fϕ

≤F (Φ

)) − F (S(t

)ϕ)+ F (S(t

)ϕ) < 1+M

(3.18)

for all n ≥ n

,i∈{1,...,N



1+h

˜γ

+2h



i−1

− ϕ

+2h

ϕ

− ϕ

 =

i−1

−ϕ

(3.19)

+ h



2 −

˜γ

1+h

˜γ



i−1

− ϕ

−ϕ

−ϕ

)+h



4 −

˜γ

1+h

˜γ



ϕ

− ϕ





≤

i−1

− ϕ

+ Ch

−ϕ

i−1

+ h





−

˜γ

1+η

˜γ



ϕ

− ϕ



for a positive constant C (independent of n ≥ n

, and i ∈{1,...,N

}).

− ϕ

i−1

= Φ

) −Φ

i−1

) (3.20)

≤Φ

) −S(t

)ϕ+ S(t

)ϕ −S(t

i−1

)ϕ+ S(t

i−1

)ϕ − Φ

i−1

)

≤ 2η

+ ρ

(η

604 W.M. Ruess

where, for η>0,ρ

(η):=sup{S(t)ϕ − S(s)ϕ|s, t ∈ [0,T

+1], |t − s|≤η}.

(Note that, as S(·)ϕ is uniformly continuous on [0,T

+1],ρ

(η

) → 0asn →∞.)

Invoking these estimates in (3.17), leads to

ϕ

− ϕ

−

i−1

−ϕ

(3.21)

≤ h





−

˜γ

1+η

˜γ



ϕ

− ϕ

 + C

ψ

 + C

(η

+ ρ

(η

))

+2h

(η(h

; ϕ

i−1

(0) + h

Fϕ

)+η(h

; x

+ h

Fϕ

))

for all n ≥ n

,i∈{1,...,N

}, and suitable positive constants C

and C

Step 3: Given 0 <s<t<T

, choose n

≥ n

large enough so that η

min {s,

(t−s),T

−t}. Then, reinvoking the upper indices, for every n ≥ n

there

exist indices 2 ≤ l ≤ k ≤ N

such that s ∈ (t

l−1

], and t ∈ (t

k−1

]. Summing

(3.21) from i =(l +1) to i = k (and taking into account the properties (3.4)–(3.6)

of the discrete-scheme approximations) leads to

Φ

(t) − ϕ

−Φ

(s) − ϕ

 = ϕ

−ϕ

−ϕ

−ϕ

 (3.22)

≤





−

˜γ

1+η

˜γ



l+1

ϕ

−ϕ

 + C

+ C

+1)(η

+ ρ

(η

))

l+1

(η(h

; ϕ

i−1

(0) + h

Fϕ

)+η(h

; x

+ h

Fϕ

))

At this point, ﬁrst note that h

ϕ

− ϕ

 =



i−1

Φ

(τ) − ϕ

 dτ, so that, by

the choice of 





−

˜γ

1+η

˜γ



l+1

ϕ

−ϕ

 (3.23)





−

˜γ

1+η

˜γ





Φ

(τ) − ϕ

 dτ −→ −γ



S(τ)ϕ − ϕ

 dτ

as n →∞.

Next, with regard to the last term on the right of (3.22), we will show that

lim

n→∞

(η(h

; ϕ

i−1

(0) + h

Fϕ

)+η(h

; x

+ h

Fϕ

)) = 0 (3.24)

uniformly over i ∈{1,...,N

l+1

≤ (T

+1), this will show that the last term on the right of (3.22)

tends to zero as n →∞.

Proof of (3.24): The result for the second term follows directly from the assump-

tions on the function η, and the fact that 0 ≤ h

≤ η

→ 0, and x

+ h

Fϕ

− x



≤ η

Fϕ

→0asn →∞uniformly over i ∈{1,...,N

As for the ﬁrst term, assume that (η(h

; ϕ

i−1

(0) + h

Fϕ

))

does

not converge to zero uniformly over i ∈{1,...,N

}. Then there exist β>0,

Linearized Stability 605

and sequences (n

)

⊂ N,n

→∞, and (i

)

∈{1,...,N

}, such that, with

:= h

, and z

:= ϕ

−1

(0) + h

−1

(0) + h

Fϕ

−1

η(λ

; z

) ≥ β for all k ∈ N. (3.25)

We now notice that, for all n ∈ N, and all i ∈{1,...,N

}, and with u

denoting

the mild solution to (PFDE) according to (S3) above,

i−1

(0) + h

Fϕ

−u

i−1

)

≤

i−1

)(0) − (S(t

i−1

)ϕ)(0)

+ h

ψ

 + h

Fϕ

≤η

(2 + M

Thus, {ϕ

i−1

(0)+h

Fϕ

| i ∈{1,...,N

}} ⊂ u

[0,T

]+η

(2+M

(with B

denoting the closed unit ball of X).

In particular, z

= x

+ y

,k∈ N, for sequences (x

)

⊂ u

[0,T

], and

)

⊂ η

(2 + M

. As u

[0,T

] is compact, there exist x

∈ X, and a

subsequence (x

)

of (x

)

such that x

→ x

. Altogether, we have λ

→ 0, and

→ x

, so that, by the assumptions on the function η, η(λ

; z

) → 0. This

contradiction to (3.25) completes the proof of (3.24).

With (3.23) and (3.24) in place, we conclude from (3.22), by letting n →∞,

S(t)ϕ − ϕ

−S(s)ϕ −ϕ

≤−γ



S(τ)ϕ − ϕ

 dτ for all 0 <s≤ t<T

By continuity in s, t on either side, this holds for all 0 ≤ s ≤ t ≤ T

. Altogether,

we thus conclude that

S(t)ϕ − ϕ

≤e

−γ

ϕ − ϕ

 for all 0 ≤ t ≤ T

,ϕ∈

E,ϕ −ϕ

 <δ. (3.26)

At this point, let ϕ ∈

E,ϕ − ϕ

 <δ,and let T

=sup{t>0 |S(s)ϕ − ϕ

≤

−γ

ϕ −ϕ

 for all 0 ≤ s ≤ t}. From (3.26), we know that T

≥ T

. Assum-

ing that T

< ∞, there exists T

≤ T

+ T

such that S(t

)ϕ −ϕ

 >

−γ

ϕ − ϕ

. However, S(T

)ϕ − ϕ

≤e

−γ

ϕ − ϕ

 <δ,so that, as

0 ≤ t

−T

≤ T

, according to (3.26), S(t

)ϕ − ϕ

 = S(t

− T

)S(T

)ϕ − ϕ

≤

−γ

−T

)

S(T

)ϕ − ϕ

≤e

−γ

ϕ − ϕ

.

This contradiction serves to complete the proof of Theorem 2.4. 

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Wolfgang M. Ruess

Fakult¨at f¨ur Mathematik

Universit¨at Duisburg-Essen

D-45117 Essen, Germany

e-mail: wolfgang.ruess@uni-due.de