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

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588 F. Ragnedda, S. Vernier Piro and V. Vespri

Proof. The functional

F (x, τ, ∇w(x, τ)) =



H(x, τ, ∇w) dx −

2(2 −p)



(x, τ ) dx

is monotone decreasing in time. In fact

dτ



Hdx=



∂H

∂˜s

∇

∂H

∂τ

≤



∂H

∂˜s

∇

= −



(div

A) w

= −



)

dx +

2 − p



dx.

Then

dτ

≤−



)

≤ 0.

As the functional F is bounded from below, this implies that, up to a subsequence,

dτ

(and therefore



(x)dx) converges to zero.

Then w converges to its limit v ∈ W

1,p

∩ L

(Ω) and v is the solution of

(5.11). 

References

[1] J.G. Berryman, C.J. Holland, Stability of the separable solution for fast diﬀusion

Arch. Rational Mech. Anal. 74 (1980), no. 4, 379–388.

[2] E. Di Benedetto, Degenerate parabolic equations, Springer Verlag, (1993)

[3]E.DiBenedetto,Y.C.Kwong,V.Vespri,Local space analyticity of solutions of

certain parabolic equations Indiana Univ. Math. J. 40 (1991), 741–765.

[4] E. Di Benedetto, J.M. Urbano, V. Vespri, Current Issues on Singular and Degenerate

Evolution Equations Evolutionary equations. Vol. 1, Handb. Diﬀer. Equ., North-

Holland, Amsterdam, (2004), 169–286.

[5]E.DiBenedetto,U.Gianazza,V.Vespri,Forward, Backward and Elliptic Harnack

Inequalities for Non-Negative Solutions to Certain Singular Parabolic Partial Diﬀer-

ential Equations Annali Scuola Nor. Sup., IX, issue 2, (2010), 385–422.

[6] U. Gianazza, V. Vespri, A Harnack Inequality for a Degenerate Parabolic Equation,

Journal of Evolution Equations 6 (2006), 247–267.

[7] J.J. Manfredi, V. Vespri, Large time behaviour of solutions to a class of Doubly

Nonlinear Parabolic equations Electronic J. Diﬀ. Eq. 2 (1994), 1–17.

[8] F. Ragnedda, S. Vernier Piro, V. Vespri, Large time behaviour of solutions to a class

of nonautonomus degenerate parabolic equations Math. Annalen 348 (2010) No. 4,

779–795.

Asymptotic Proﬁle 589

[9]F.Ragnedda,S.VernierPiro,V.Vespri, Asymptotic time behaviour for non-

autonomous degenerate parabolic problems with forcing term Nonlinear Anal. 71

(2009), e2316–e2321.

[10] G. Savar´e, V. Vespri, The asymptotic proﬁle of solutions of a class of doubly nonlinear

equations Nonlinear Anal. 22 (1994), no. 12, 1553–1565.

[11] J.L. V´azquez, Asymptotic behaviour and propagation properties of the one-dimen-

sional ﬂow of gas in a porous medium Trans. Amer. Math. Soc., 277 (1983), no. 2,

507–527.

[12] J.L. V´azquez, Smoothing and decay estimates for nonlinear diﬀusion equations.

Equations of porous medium type Oxford Lecture Series in Mathematics and its

Applications, 33, Oxford University Press, Oxford, 2006.

F. Ragnedda and S. Vernier Piro

Dipartimento di Matematica e Informatica

Viale Merello 92

I-09123 Cagliari, Italy

e-mail: ragnedda@unica.it

svernier@unica.it

V. Vespri

Dipartimento di Matematica “U. Dini”,

viale Morgagni 67/a

I-50134 Firenze, Italy

e-mail: vespri@math.unifi.it

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 591–607

 2011 Springer Basel AG

Linearized Stability for Nonlinear Partial

Diﬀerential Delay Equations

Wolfgang M. Ruess

Dedicated to Herbert Amann on the oc casion of his 70th birthday

Abstract. The object of the paper are partial diﬀerential delay equations of

the form ˙u(t)+Bu(t)  F (u

),t≥ 0,u

= ϕ,withB ⊂ X × Xω-accretive

in a Banach space X. We extend the principle of linearized stability around

an equilibrium from the semilinear case, with B linear, to the fully nonlinear

case, with B having a linear ‘resolvent-diﬀerential’ at the equilibrium.

Mathematics Subject Classiﬁcation (2000). Primary 35R10, 47J35; Secondary

47H06, 47H20.

Keywords. Nonlinear partial diﬀerential delay equations; accretive operators;

linearized stability; subtangential conditions.

1. Introduction

The object of study of this paper is the following partial functional diﬀerential

equation with delay:

(PFDE)



˙u(t)+Bu(t)  F (u

),t≥ 0

= ϕ ∈

B ⊂ X ×X is a (generally) nonlinear and multivalued operator in a Banach (state)

space X, with (B + ωI) accretive, some ω ∈ R , and, for given I =[−R, 0], R>0

(ﬁnite delay), or I =(−∞, 0] (inﬁnite delay), and t ≥ 0,u

: I → X is the history

of u up to t : u

(s)=u(t + s),s∈ I, and ϕ : I → X is a given initial history out

of a subset

E of a space E of continuous functions from I to X. Moreover, F is a

given history-responsive operator with domain D(F ) ⊂ E and range in X.

Our objective is a solution to the following problem of linearized stability for

(PFDE): Assume that there exists an equilibrium solution ϕ

∈ E to (PFDE), let

592 W.M. Ruess

= ϕ

(0), and assume that there exist ‘linearizations’

B of B at x

, and

F of F

at ϕ

such that the solution semigroup (in E) to the ‘linearization’

(PFDE)

lin



˙u(t)+

Bu(t) 

F (u

),t≥ 0

= ϕ ∈ E

of (PFDE) is exponentially stable. Is it then true that the equilibrium ϕ

is locally

exponentially stable with respect to the nonlinear problem (PFDE)?

The existing positive results in this direction generally are for the semilinear

case, with B : D(B) ⊂ X → X linear and single-valued, and either the inﬁnitesimal

generator of a C

-semigroup of bounded linear operators on X, or a Hille-Yosida

operator, and under the assumption that F be globally Fr´echet-diﬀerentiable with



: E → B(E,X) being locally Lipschitz, or just Fr´echet-diﬀerentiable at ϕ

(c.f.

[1] [15, Thm. 6.1], [19, 20, 27]). For more general B ⊂ X × X linear, and F not

globally, but possibly deﬁned only on ‘thin’ subsets of E, see[25,Thm.4.2].For

general, possibly nonlinear B ⊂ X ×X, and under local range conditions on both

B and F (possibly deﬁned only on ‘thin’ subsets of E), a positive answer has been

given in [24, Thm. 4.1].

The purpose of this paper is to extend all of these results in two directions:

(a) the fully nonlinear case with B ⊂ X × X nonlinear, ω-accretive, and linearly

‘resolvent-diﬀerentiable’ at x

(Deﬁnition 2.2), and (b) the local case of F possibly

being deﬁned only on a ‘thin’ subset of E, and not necessarily locally Lipschitz,

and under a subtangential condition on B and F as in [25].

As has been demonstrated by the applications in [25, Section 5], the generality

in having F only deﬁned on ‘thin’ subsets (possibly not even containing an interior

point), and not necessarily locally Lipschitz, is crucial for population models set

up in the (natural) state space L

(Ω), Ω an open subset of R

For existence results for (PFDE) in general, mostly in the global context, we

refer here to [4, 5, 6, 7, 12, 13, 14, 16, 21, 23, 26, 28, 29, 30]; for a much more

complete list, the reader is referred to the survey [22]. The monograph [31] surveys

the semilinear case with B the generator of a C

-semigroup.

Notation and terminology. Throughout this paper, all Banach spaces are assumed

to be over the reals. For X a Banach space, and λ ∈ R \{0}, deﬁne [·, ·]

: X ×X →

R by

[x, y]

x + λy−x

Then, as, for ﬁxed x, y ∈ X, the function {λ → [x, y]

} is nondecreasing for λ>0,

one can deﬁne the bracket [x, y] (the right-hand Gˆateaux-derivative of the norm

at x in the direction of y)by

[x, y] = lim

λ0

[x, y]

=inf

λ>0

[x, y]

(cf. [3, 18]).

B(X) denotes the space of all bounded linear operators from X into X. Given

a subset D of X, cl D will denote its closure in X. Recall that a subset C ⊂ X ×X

Linearized Stability 593

is said to be accretive in X if for each λ>0, and each pair (x

) ∈ C, i ∈{1, 2},

we have (x

+ λy

) −(x

+ λy

)≥x

− x

, and ω-accretive in X for some

ω ∈ R , if (C + ωI) is accretive. If, in addition, R(I + λC)=X for all λ>0with

λω < 1, then C is said to be m−ω-accretive. If C ⊂ X × X is ω-accretive, then,

for any λ>0withλω < 1,J

=(I + λC)

−1

denotes the resolvent of C.

Notions of solutions. Given an initial-history space E of functions from I to X, T >

0, and B ⊂ X × Xω-accretive for some ω ∈ R, a function u : I ∪ [0,T] → X will

be called a mild solution to (PFDE) if (i) u

= ϕ, and (ii) u

|[0,T]

is continuous

and is a mild solution to the initial value problem

(CP)



˙u(t)+Bu(t)  f(t), 0 ≤ t ≤ T,

u(0) = ϕ(0)

with f(t)=F (u

). In turn, given f ∈ L

(0,T; X), a continuous function u :

[0,T] → X with u(0) = ϕ(0) is a mild solution to (CP) if, given any >0, there ex-

ists an -discrete-scheme approximation D

,...,t

; u

,...,u

; f

,...,f

)con-

sisting of an -partition of the interval [0,T],

0 ≤ t

< ···<t

≤ T, t

<,T− t

<, t

− t

i−1

<,i∈{1,...,N},

(1.1)

and elements {u

,...,u

}, and {f

,...,f

} in X such that

∈ D(B),i ∈{1,...,N}, (1.2)

−u

i−1

−t

i−1

+ Bu

 f

,i∈{1,...,N},



i−1

f

− f(τ)dτ < , u

− u(0) <,and

such that, if the step function u



:[t

] → X is deﬁned by



(t)=



t = t

t ∈ (t

i−1

],i∈{1,...,N},

then u



(t) − u(t) <uniformly over t ∈ [t

For all these notions and the general theory of accretive sets and evolution

equations, the reader is referred to [3, 18].

2. Linearized stability for (PFDE)

We shall place our results in the context of general local subtangential conditions

on B and F for the existence of solutions to (PFDE), as well as in the context of

a general class of initial-history spaces E as in [25].

2.A The initial history space. Given a Banach (state) space X, and letting I =

(−∞, 0] (inﬁnite delay), or I =[−R, 0] for some R>0 (ﬁnite delay), the initial

history space is assumed to be a Banach space (E, ·) of continuous functions

ϕ : I → X with the following properties:

594 W.M. Ruess

(E1) (a) For all ϕ ∈ E, ϕ(0)≤ϕ.

(b) For all x ∈ X, ˜x ∈ E, where ˜x(s) ≡ x, s ∈ I, and

≥ 1 such that ˜x≤C

x for all x ∈ X.

(d) For ϕ, (ϕ

)

in E, if ϕ

− ϕ→0, then (1) ϕ

(s) − ϕ(s)→0for

all s ∈ I, and (2)



(s)ds →



ϕ(s)ds for all α, β ∈ I, α < β.

(E2) If λ>0,x∈ X, ψ ∈ E, and ϕ

λ,x

∈ C

(I; X) is the solution to ϕ − λϕ



ψ, with ϕ(0) = x, then ϕ

λ,x

∈ E, and

λ,x

≤ max{x, ψ}.

(E3) (a) If x : I ∪ [0, ∞) → X is continuous, and x

∈ E, then

(i) x

∈ E for all t ≥ 0, and

(ii) the map {t → x

} is continuous from R

into E.

(b) There exist M

≥ 1, and a locally bounded function M

: R

→ R

such that, given x : I ∪[0, ∞) → X as in (a) above,

x

≤M

x

 + M

(t)max

0≤s≤t

x(s) for all t ≥ 0.

(Concerning these axioms for E, compare [9, 11].)

Examples. In the ﬁnite-delay case, usually, the initial-history space will be E =

C([−R, 0]; X) with sup-norm. For the inﬁnite-delay case I =(−∞, 0], the most

prominent initial-history spaces are weighted sup-norm spaces of the type E

{ϕ ∈ C(R

−

,X):vϕ ∈ BUC(R

−

; X)}, with norm ϕ

:= sup{v(s) ϕ(s)|s ∈

−

}, where the (weight-) function v : R

−

→ (0, 1] has the following properties:

(v1) v is continuous, nondecreasing, and v(0) = 1;

(v2) lim

u→0

− v(s + u)/v(s) = 1 uniformly over s ∈ R

−

Typical such weight functions are v(s) ≡ 1(with,inthiscase,E

= BUC(R

−

,X)

with sup-norm), v(s)=e

μs

, or v(s)=(1+|s| )

−μ

,μ>0 (spaces of ‘fading memory

type’). (The Banach spaces E

are sometimes called UC

-spaces, v =1/g, and have

been considered by various authors, cf. [11].)

Aside from E

, also the following subspaces fulﬁll axioms (E1)–(E3):

(a) E

= {ϕ ∈ E

| lim

s→−∞

v(s)ϕ(s)exists}, and

(b) E

= {ϕ ∈ E

| lim

s→−∞

v(s)ϕ(s)=0}, in case lim

s→−∞

v(s)=0.

2.B The framework for (PFDE). Givenaninitial-historyspaceE as in 2.A,we

start from the following assumptions:

(A1) B ⊂ X × X is ω-accretive for some ω ∈ R .

(A2)

X ⊂ X is a closed subset of X with

X ∩ D(B) = ∅.

(A3) (a)

= {ϕ ∈ E | ϕ(0) ∈ cl (D(B) ∩

X)}.

(b)

E ⊂

is a closed subset of

X ∩ D(B),ψ ∈

E, and λ>0 is suﬃciently small, then

λ,x

∈

E, where ϕ

λ,x

is the solution ϕ ∈ E to ϕ − λϕ



= ψ, ϕ(0) = x.

(A4) The operator F :

→ X is such that

(a) F :

→ X is continuous;

(b) there exists L

> 0 such that, if ϕ

,ϕ

∈

with ϕ

(0) = ϕ

(0),

then Fϕ

− Fϕ

≤L

ϕ

− ϕ

, and

Linearized Stability 595

> 0 such that, if ϕ

,ϕ

∈

with ϕ

−ϕ

 =

ϕ

(0) − ϕ

(0), then [ϕ

(0) − ϕ

(0),Fϕ

− Fϕ

] ≤ M

ϕ

− ϕ

.

2.C Solution theory for (PFDE). As for the existence of solutions to (PFDE), we

recall the main existence result of [25, Thm. 2.1]:

Theorem 2.1. Given the assumptions (A1)–(A4),if

(STC) lim inf

λ→0

dist(ψ(0) + λF (ψ); (I + λB)(D(B) ∩

X)) = 0 for all ψ ∈

then we have:

(a) For al l ϕ ∈

E there exists a global mild solution u

(PFDE)



˙u(t)+Bu(t)  F (u

),t≥ 0,

= ϕ

with (u

)

∈

E for all t ≥ 0. In particular, u

(t) ∈

X for all t ≥ 0.

(b) If, in addition, either F |

E → X is locally Lipschitz-continuous on

or there exists M

> 0 such that [ϕ(0) − ψ(0),Fϕ− Fψ] ≤ M

ϕ −ψ for

all ϕ, ψ ∈

E, then, for any ϕ ∈

E, the mild solution u

as in (a) is unique

amongst all mild solutions u to (PFDE) with the property that u

∈

E for all

t ≥ 0.

For our result on linearized stability, we need the following approach to con-

structing the mild solution u

Associate with (PFDE) the operator A in E deﬁned by



D(A)={ϕ ∈

| ϕ



∈ E,ϕ(0) ∈ D(B),ϕ



(0) ∈ Fϕ− Bϕ(0)}

Aϕ := −ϕ



,ϕ∈ D(A).

(2.1)

Given the assumptions of Theorem 2.1, the following assertions hold ([25]):

(S1) The operator A is γ-accretive in E with γ = max{0,ω+ M

(S2) For every ϕ ∈

E, there exists a unique mild solution Φ

: R

→ E to the

Cauchy problem in E corresponding to A,

(CP)(A; ϕ;0)



Φ(t)+AΦ(t)=0,t≥ 0,

Φ(0) = ϕ.

The solution semigroup (S(t))

t≥0

generated by −A via S(t)ϕ := Φ

(t)leaves

E invariant.

(S3) If ϕ ∈

E, and u

: I ∪R

→ X is deﬁned by

(t)=



ϕ(t) t ∈ I

(S(t)ϕ)(0) t ≥ 0,

(2.2)

then S(t)ϕ =(u

)

for t ≥ 0 (i.e., (S(t))

t≥0

acts as a translation), and the

function u

is a global mild solution to (PFDE).

596 W.M. Ruess

2.D Resolvent diﬀerentiability. The following concept of diﬀerentiating a (possibly)

multivalued operator will be basic for our result.

Deﬁnition 2.2. For C ⊂ Z × Z an α-accretive operator in a Banach space Z,

and z

∈ Z with z

∈ R(I + λC) for all λ>0,λα<1, an ˜α-accretive operator

C ⊂ Z × Z with (R(I + λC) − z

) ⊂ R(I + λ

C) for all λ>0 small enough, is

called a resolvent-diﬀerential of C at z

, if the following holds:

(RD)

⎧

⎪

⎨

⎪

⎩

There exists a function η :

{{λ}×R(I + λC) | 0 <λ<λ

}→R

some λ

> 0,λ

α<1, with lim

(λ,z)→(0,ˆz)

η(λ, z)=0, such that for

every >0, there exist δ>0, and λ

> 0,λ

≤ λ

, such that, if

z ∈ R(I + λC), and z − z

 <δ,then

z − J

−J

(z − z

)

≤ λ z − z

 + λη(λ, z)

for all 0 <λ≤ λ

Remark 2.3. The notion of resolvent-diﬀerentiability has originally been intro-

duced in [24]. As discussed in [24, Remarks 2.2], with regard to ω-accretive op-

erators, this notion seems to be the natural extension of Fr´echet-diﬀerentiability

from ‘single-valued’ to ‘multivalued’. In particular, while the principle of linearized

stability for the Cauchy problem ˙u(t)+Au(t)  0holdsforA single-valued and

Fr´echet-diﬀerentiable at the equilibrium, it carries over to (α-accretive) multival-

ued A if A has a linear resolvent-diﬀerential

A at the equilibrium ([24, Thm. 2.1]).

More to the point, if A : D(A) ⊂ X → X is (single-valued and) α-accretive,

with R(I + λA) ⊃ D(A)forλ>0 small enough, x

∈ D(A), and A has an F-

diﬀerential

A ∈ B(X)atx

relative to D(A)(inthesenseof(L) (3) below), then

A is a resolvent-diﬀerential of A at x

(with η(λ, x)=M

−x

, for some

constant M>0); see [24, Remark 2.3].

In order to formulate the linearization principle for (PFDE), we start from

the following additional assumptions.

(L) (1) There exists an equilibrium solution ϕ

∈ D(A) ∩

E of the solution semi-

group (S(t))

t≥0

for (PFDE) such that x

:= ϕ

(0) ∈ R(I + λB) for all λ>0,

λω < 1.

(2) There exists a linear and m−˜ω-accretive resolvent-diﬀerential

B ⊂ X ×X of

B at x

, some ˜ω ∈ R .

(3) There exists a bounded linear operator

F : E → X that is a D(A)-Fr´echet-

derivative of F at ϕ

, i.e., given any >0, there exists δ>0 such that, if

ϕ ∈ D(A), and ϕ − ϕ

 <δ, then

Fϕ− Fϕ

−

F (ϕ − ϕ

)

≤  ϕ − ϕ

.

With assumption (L) in place, we consider in E the solution operator



A)={ϕ ∈ E | ϕ



∈ E,ϕ(0) ∈ D(

B),ϕ



(0) ∈

F (ϕ) −

Bϕ(0)}

Aϕ := −ϕ



,ϕ∈ D(

A),

associated with (PFDE)

lin

Linearized Stability 597

We are now able to formulate the main result on linearized stability for

(PFDE).

Theorem 2.4. Given the assumptions (L) and those of Theorem 2.1,ifthereexists

˜γ>0 such that the linearized operator (

A − ˜γI) ⊂ E × E is accretive, then the

initial history problem (PFDE) is locally exponentially stable at the equilibrium ϕ

in the following sense: given any 0 <γ

< ˜γ, there exists δ>0 such that, for all

ϕ ∈

E with ϕ − ϕ

 <δ,

(t) − x

≤

)

−ϕ

≤ e

−γ

ϕ −ϕ

 for all t ≥ 0,

where u

: R

→ X is the mild solution to (PFDE) as in Theorem 2.1.

Remarks 2.5. 1. The main improvement of Theorem 2.4 over existing results is the

general fully nonlinear and local context of assumptions (A1)–(A4),asopposed

to having B m−ω-accretive, and F globally deﬁned, in tandem with the facts (a)

that B need not be linear, but just linearly resolvent-diﬀerentiable at x

, and (b)

that F need not be Fr´echet-diﬀerentiable, but only D(A)-diﬀerentiable at ϕ

as in

assumption (L), (3). In this regard, notice that, if B ⊂ X × X is m−ω-accretive,

and F is globally deﬁned and Lipschitz, then (A1)–(A4) and condition (STC) are

fulﬁlled automatically for

X = X, and

E =

2. In particular, we note that in the global case

X = X,

E =

,andF : E → X

globally Lipschitz, versions of Theorem 2.4 have been proved (a) in the ﬁnite-

delay case for B : D(B) ⊂ X → X linear and single-valued, and generating

a C

-semigroup, under the assumption that F be globally Fr´echet-diﬀerentiable

with F



: E → B(E,X) being locally Lipschitz ([15, Thm. 6.1], [19, 20]), and (b) in

the inﬁnite-delay case for B : D(B) ⊂ X → X (single-valued and) a Hille-Yosida

operator, and F Fr´echet-diﬀerentiable at the equilibrium ([1]). (Note that Hille-

Yosida operators are – single-valued, linear and – m−ω-accretive for an equivalent

norm on X.)

3. Under local range ﬂow-invariance conditions for

X ⊂ X, and

E ⊂ E, more

restrictive than condition (STC) (compare [25, Lemma 2.7]), versions of Theorem

2.4 have been proved in [27, Thm. 2.4] for B : D(B) ⊂ X → X linear, and the

inﬁnitesimal generator of a C

-semigroup of bounded linear operators on X, and in

[24, Thm. 4.1] for linearly resolvent-diﬀerentiable B ⊂ X ×Xω-accretive. In both

of these instances, the corresponding proofs were based on the range condition

R(I + λA) ⊃

E for λ>0 small enough, so that, in constructing the solution

semigroup in E, we were able to work with the resolvents of A. This marks a

substantial diﬀerence to the proof for the more general situation of Theorem 2.4,

where, instead, we are forced to consider general -discrete-scheme approximations

(see Section 3).

Finally, we note that the special version of Theorem 2.4 for B =

B has been

given in [25, Thm. 4.2], by means of a much more direct method of proof than the

one for the nonlinear case in Section 3 below.