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108 T. Cie´slak and Ph. Lauren¸cot
whence
1
4
∂
y
Ψ(f(t))
2
L
2
(0,M)
≤ L
1
(0) + M
3
+ M
"
"
"
"
Ψ
1
M
"
"
"
"
+ M
2
Ψ
1
(Σ(t)). (4.6)
Using again Lemma 4.1, we have
Ψ(f(t))
L
1
(0,M)
≤ M
3/2
∂
y
Ψ(f(t))
L
2
(0,M)
+ M
"
"
"
"
Ψ
1
M
"
"
"
"
≤ 2M
3/2
L
1
(0) + M
3
+ M
"
"
"
"
Ψ
1
M
"
"
"
"
+ M
2
Ψ
1
(Σ(t))
1/2
+ M
"
"
"
"
Ψ
1
M
"
"
"
"
.
Combining the previous inequality with (4.6) and the Poincar´e inequality leads us
to the bound
Ψ(f(t))
H
1
(0,M)
≤ C
6
(T ) ,t∈ [0,T] ∩[0,T
max
) , (4.7)
for all T>0. Together with the continuous embedding of H
1
(0,M)inL
∞
(0,M),
(4.7) gives
−C
7
(T ) ≤ Ψ(f(t, y)) ≤ C
7
(T ) , (t, y) ∈ ([0,T] ∩ [0,T
max
)) × [0,M] .
Since
lim
r→0
Ψ(r)=−∞
due to a ∈ L
1
(1, ∞), the above lower bound on Ψ(f) ensures that f(t) cannot
vanish in finite time, from which Theorem 1.1 (ii) follows as already discussed in
Section 2.
Acknowledgment
This work was done while the first author held a post-doctoral position at the
University of Z¨urich.
References
[1] A. Blanchet, J.A. Carrillo, and Ph. Lauren¸cot, Critical mass for a Patlak-Keller-Segel
model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential
Equations 35 (2009), 237–242.
[2] P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating
Langevin particles in D dimensions, Phys. Rev. E 69 (2004), 016116.
[3] T. Cie´slak and Ph. Lauren¸cot, Finite time blow-up results for radially symmetric
solutions to a critical quasilinear Smoluchowski-Poisson system,C.R.Acad.Sci.
Paris S´er. I 347 (2009), 237–242.
[4] T. Cie´slak and Ph. Lauren¸cot, Looking for critical nonlinearity in the one-dim-
ensional quasilinear Smoluchowski-Poisson system, Discrete Contin. Dynam. Sys-
tems A 26 (2010), 417–430.
[5] T. Cie´slak and M. Winkler, Finite-time blow-up in a quasilinear system of chemo-
taxis, Nonlinearity 21 (2008), 1057–1076.
Global Existence vs. Blowup 109
[6] W. J¨ager and S. Luckhaus, On explosions of solutions to a system of partial differen-
tial equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824.
[7] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,Adv.
Math.Sci.Appl.5 (1995), 581–601.
[8] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic system modeling chemo-
taxis in two-dimensional domains,J.Inequal.Appl.6 (2001), 37–55.
[9] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate
Keller-Segel models, Adv. Differential Equations 12 (2007), 121–144.
[10] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-
critical cases to degenerate Keller-Segel systems, Differential Integral Equations 19
(2006), 841–876.
Tomasz Cie´slak
Institute of Applied Mathematics
Warsaw University
Banacha 2
PL-02-097 Warszawa, Poland
e-mail: T.Cieslak@impan.gov.pl
Philippe Lauren¸cot
Institut de Math´ematiques de Toulouse
CNRS UMR 5219
Universit´edeToulouse
118 route de Narbonne
F-31062 Toulouse Cedex 9, France
e-mail: Philippe.Laurencot@math.univ-toulouse.fr
Progress in Nonlinear Differential Equations
and Their Applications, Vol. 80, 111–130
c
2011 Springer Basel AG
Perturbation Results for
Multivalued Linear Operators
Ronald Cross, Angelo Favini and Yakov Yakubov
Abstract. We give some perturbation theorems for multivalued linear opera-
tors in a Banach space. Two different approaches are suggested: the resolvent
approach and the modified resolvent approach. The results allow us to handle
degenerate abstract Cauchy problems (inclusions). A very wide application of
obtained abstract results to initial boundary value problems for degenerate
parabolic (elliptic-parabolic) equations with lower-order terms is studied. In
particular, integro-differential equations have been considered too.
Mathematics Subject Classification (2010). Primary 47A06; 47A55; Secondary
35K65; 35K30; 35M13.
Keywords. Multivalued linear operators, selection, resolvent, modified resol-
vent, perturbation, parabolic equation, elliptic-parabolic equation, degenerate
equation.
1. Introduction
Degenerate evolution equations in Banach spaces and their applications to par-
tial differential equations constitute a very wide field of mathematical research.
Many different methods exist to handle this subject (see, e.g., [11]). Two different
approaches were introduced in [6]. The first one relates to multivalued linear opera-
tors, while the second uses a modified resolvent and operational method, extending
G. Da Prato and P. Grisvard’s approach (see also [7] for applications to nonlinear
equations). In whichever case, a basic role is played by the resolvent estimates of
the operators involved. A number of applications to singular linear parabolic dif-
ferential equations has been given in [4], [5], which improved the previous results in
[8] and [6]. In particular, [5] deals directly with second-order differential operators
with lower-order terms.
The second author is a member of G.N.A.M.P.A. and the I.N.D.A.M.; the third author is sup-
ported by I.N.D.A.M. and the Israel Ministry of Absorption.
112 R. Cross, A. Favini and Y. Yakubov
In this paper we obtain the relevant perturbation theorems for such equations.
To the best of our knowledge, such perturbation results have not appeared in the
existing literature.
More precisely, Section 2 describes the resolvent approach of perturbing a
multivalued linear operator by another (possibly) multivalued linear operator and
satisfying some estimate. This goal is reached by some arguments from [1]. Section
3 deals with the modified resolvent approach which is, on the other hand, strictly
related to the first type. Section 4 furnishes a large number of concrete examples
from partial differential equations to which the developed abstract theory applies.
We note that in Examples 5–7, an alternative approach to certain equations of [5],
related to gradient estimates, is indicated.
2. Resolvent approach
Denote by X, Y Banach spaces, by L(X, Y ) a space of all bounded linear operators
from X into Y ,byML(X, Y ) a space of all multivalued linear operators from X
into Y ,byA, B multivalued linear operators from ML(X, Y ). If X = Y then
denote L(X):=L(X, Y )andM L(X):=M L(X, Y ). The norm of Au is defined
as follows
Au := inf{y
Y
: y ∈ Au}, ∀u ∈ D(A),
and
A
ML(X,Y )
:= sup
u
X
≤1
Au
Y
.
By A
0
we denote a selection (or a single-valued linear part) of A, i.e.,
A = A
0
+ A − A, D(A
0
)=D(A).
Obviously,
Au = A
0
u + A(0), ∀u ∈ D(A).
Here and everywhere, A(0) stands for A0.
Definition 2.1. Given an operator A ∈ ML(X). By ρ(A) ⊂ C we denote the
resolvent set of A, i.e., λ ∈ ρ(A) if and only if the inverse operator (λI −A)
−1
is a
linear bounded single-valued operator defined on the whole space X.Inthiscase
we denote R(λ, A):=(λI − A)
−1
and call it the resolvent of A.
Let E be a linear subspace of X.DenotebyP
E
the multivalued linear projec-
tion given by P
E
x := E for any x ∈ X. Thus the graph of P
E
, G(P
E
)=X ×E.Ob-
viously, the kernel of P
E
, N(P
E
)=X and P
E
(0) = E. Next, define I
E
∈ M L(X)
by I
E
:= I + P
E
. Then, we have a series of simple facts: N (I
E
)=E, the image
R(I
E
)=X, I
E
(0) = E, G(I
E
)={(x, x + e):x ∈ X, e ∈ E}, I
E
x = x + E, for all
x ∈ X, I
E
=sup
x=0
I
E
x
x
=sup
x=0
x+E
x
≤ 1(sinceI
E
x = x+ E =inf{x+y :
Perturbation Results for Multivalued Linear Operators 113
y ∈ E}), and the minimum modulus (cf. [1, II.2.2])
γ(I
E
)=
∞, if E is dense in X,
inf
x/∈E
x+E
x+E
, otherwise
=
∞, if
E = X,
1, if E = X.
Proposition 2.2. Let S ∈ M L(X) satisfy D(S)=X and S < 1.Then,the
operator I − S has a dense range.
Proof. Write E := S(0) (it is known, [1, I.2.4], that S(0) is a linear subspace of
X). Since γ(I
E
) > 0, I
E
is open (see [1, II.3.2(b)]). Also, R(I
E
)=X, S(0) = E =
I
E
(0), and S < 1 ≤ γ(I
E
). Hence, by [1, III.7.5], I
E
−S has a dense range. Since
I − S = I
E
−S, the result follows.
Obviously, if X is finite dimensional, then D((I − S)
−1
)=X.
Proposition 2.3. Let S ∈ ML(X) satisfy D(S)=X and S < 1,andletS(0) be
closed. Then, (I − S)
−1
∈ M L(X) is continuous if and only if R(I −S) is closed.
Proof. By Proposition 2.2, if R(I −S)isclosedthenI −S is surjective, therefore,
D((I − S)
−1
)=X which implies, by the closed graph theorem (see [1, III.4.2]),
that (I −S)
−1
is continuous. Conversely, if (I − S)
−1
is continuous then I −S is
open (see [1, II.3.1]) and, by [1, III.4.2(b)], R(I − S)isclosed.
Let us now prove the main proposition.
Proposition 2.4. Let S ∈ ML(X) satisfy D(S)=X and S < 1,andletS(0) be
closed. Then, (I − S)
−1
∈ ML(X) is everywhere defined and continuous.
Proof. In order to prove that the operator (I − S)
−1
is continuous, it is enough,
by Proposition 2.3, to show that R(I − S) is closed. By [1, III.4.4], R(I − S)is
closed if and only if the range of the adjoint R((I −S)
) is closed. By [1, III.1.5(b)],
(I − S)
= I −S
. Thus, we are going to show that R(I −S
)isclosed.
By [1, II.3.2(a)] and [1, III.1.13], the adjoint operator S
is continuous. Then,
by [1, III.4.2(a)], D(S
) is a closed subspace of X
.SinceD(S)=X then, by [1,
III.1.4(b)], S
is single-valued. By [1, III.1.13], S
≤S < 1. Denote S
=
1 − d,where0<d<1. Some trivial calculations show that the operator I − S
is injective. Indeed, let (I −S
)x
= 0. Then, 0 = (I − S
)x
≥x
−S
x
≥
x
−S
x
= dx
≥0, i.e., x
= 0. Let us now show that I −S
is open. By
[1, II.2.1], since I −S
is injective, γ(I −S
)= inf
x
=0,x
∈D(S
)
(I−S
)x
x
≥ d>0. This
proves, by [1, II.3.2(b)], that I −S
is open as a map from D(I −S
)=D(S
)onto
R(I − S
), i.e., the inverse (I − S
)
−1
is continuous, by [1, II.3.1] and, therefore,
D(S
)andR(I −S
) are isomorphic. This implies, in turn, that R(I −S
)isclosed
since D(S
) is closed. Therefore, as mentioned above, R(I − S) is closed and, by
Proposition 2.3, the operator (I − S)
−1
is continuous.
Combining Proposition 2.2 with the fact that R(I −S) is closed, we conclude
that R(I −S)=X.
114 R. Cross, A. Favini and Y. Yakubov
Note that for I − S to be only surjective one should not claim that S(0) is
closed or S < 1. Let us give some other sufficient conditions for that.
Proposition 2.5. If S ∈ ML(X)(with D(S)=X) has a continuous selection S
0
such that S
0
< 1 then I −S is surjective.
Proof. If S
0
< 1thenI − S
0
is surjective (from the single-valued theory). On
the other hand, I − S = I − S
0
+ S − S, i.e., (I − S)x =(I − S
0
)x + S(0) for all
x ∈ X. Hence, R(I − S) ⊃ R(I − S
0
)=X.
Proposition 2.6. If S ∈ ML(X)(with D(S)=X), S < 1,andthereexistsa
linear projection, with the norm equal to 1, defined on R(S) with kernel S(0),then
I − S is surjective.
Proof. Let P be the norm-1 projection and let S
0
= PS. Then, by [1, II.3.13],
S
0
= PS≤P S = S < 1. Now apply Proposition 2.5.
Note that if X is a Hilbert space then there exists a norm-1 linear projection
defined on R(S)withkernelS(0).
We now pass to the main perturbation theorems.
Theorem 2.7. Let the following conditions be satisfied:
1. A ∈ M L(X) and, for some η ∈ (0, 1],
R(λ, A)≤C|λ|
−η
, for sufficiently large λ ∈ Γ,
where Γ is an unbounded set of the complex plane;
2. B ∈ ML(X), D(B) ⊃ D(A), B(0) is closed, and, for any ε>0,thereexists
C(ε) > 0 such that
Bu≤εAu
η
u
1−η
+ C(ε)u, ∀u ∈ D(A).
Then, for every sufficiently large λ ∈ Γ, the multivalued linear operator (λI −A −
B)
−1
is defined on the whole space X and
(λI − A − B)
−1
≤C|λ|
−η
, for sufficiently large λ ∈ Γ.
Proof. By, e.g., [6, Theorem 1.7, p. 24], −I + λR(λ, A) ⊂ AR(λ, A), ∀λ ∈ ρ(A).
Then, for sufficiently large λ ∈ Γ, by condition (1),
AR(λ, A)≤1+|λ|R(λ, A)≤C|λ|
1−η
.
Hence, by conditions 1 and 2, for sufficiently large λ ∈ Γandforanyv ∈ X,
BR(λ, A)v≤εAR(λ, A)v
η
R(λ, A)v
1−η
+ C(ε)R(λ, A)v
≤ (εC + C(ε)|λ|
−η
)v.
Therefore,
BR(λ, A)≤q<1, for sufficiently large λ ∈ Γ.
Perturbation Results for Multivalued Linear Operators 115
Then, by Proposition 2.4, the multivalued linear operator (I −BR(λ, A))
−1
is ev-
erywhere defined and continuous for the same λ, i.e., by [1, II.3.2(a)], the operator
has a bounded norm
(I − BR(λ, A))
−1
≤C, for sufficiently large λ ∈ Γ. (2.1)
Further, since λI −A is injective (see, e.g., [1, Exercise VI.1.2, p. 220]) then,
by, e.g., [1, formula I.1.3/(9), p. 3] and [9, Proposition A.1.1/(e), p. 281], on D(A),
(I −BR(λ, A))(λI − A) ⊂ λI −A −BR(λ, A)(λI −A)=λI − A − B. (2.2)
Then, by the definition of the inverse operator of multivalued linear operators (see,
e.g., [1, p. 1])
[(I − BR(λ, A))(λI − A)]
−1
⊂ (λI − A − B)
−1
.
From this, taking into account, e.g., [9, Proposition A.1.1/(a), p. 280], we get, for
sufficiently large λ ∈ Γ,
R(λ, A)(I −BR(λ, A))
−1
⊂ (λI − A − B)
−1
. (2.3)
In fact, the left-hand side operator is defined on the whole space X,whichmeans
that the image of the left-hand side operator in (2.2) is X, i.e., also the image
R(λI −A −B)=X. This says that the operator (λI −A −B)
−1
is defined on the
whole space X for sufficiently large λ ∈ Γ. On the other hand, by (2.1), (2.3), [1,
Corollary II.3.13, p. 38], and condition 1, we get
(λI − A − B)
−1
≤R(λ, A)(I − BR(λ, A))
−1
≤ C|λ|
−η
, for sufficiently large λ ∈ Γ.
Conditions of Theorem 2.7 do not guarantee that the operator (λI−A−B)
−1
is single valued, that is why we could not say that (λI −A −B)
−1
= R(λ, A + B).
Let us now formulate the results which state that (λI −A −B)
−1
= R(λ, A + B).
Theorem 2.8. Let the following conditions be satisfied:
1. A ∈ M L(X) and, for some η ∈ (0, 1],
R(λ, A)≤C|λ|
−η
, for sufficiently large λ ∈ Γ,
where Γ is an unbounded set of the complex plane;
2. B ∈ ML(X), D(B) ⊃ D(A), B(0) is closed, and, for any ε>0,thereexists
C(ε) > 0 such that
Bu≤εAu
η
u
1−η
+ C(ε)u, ∀u ∈ D(A);
3. For sufficiently large λ ∈ Γ, the operator (I −R(λ, A)B)
−1
is single valued.
Then, every sufficiently large λ ∈ Γ belongs to ρ(A + B) and
R(λ, A + B)≤C|λ|
−η
, for sufficiently large λ ∈ Γ.
116 R. Cross, A. Favini and Y. Yakubov
Proof. From I ⊂ (λI − A)R(λ, A) and [9, Proposition A.1.1/(d) and (e), pp.280-
281], for sufficiently large λ ∈ Γ,
λI − A − B ⊂ λI − A − (λI − A)R(λ, A)B ⊂ (λI − A)(I −R(λ, A)B).
Then, by the definition of the inverse operator of multivalued linear operators (see,
e.g., [1, p.1]) and, e.g., [9, Proposition A.1.1/(a), p.280],
(λI − A − B)
−1
⊂ (I − R(λ, A)B)
−1
R(λ, A), (2.4)
for the same λ. Since the right-hand side operator is single valued, for sufficiently
large λ ∈ Γ, then (λI −A − B)
−1
is also single valued for the same λ. Combining
this with the result of Theorem 2.7, we get that these λ belong to ρ(A + B)and
(λI − A − B)
−1
= R(λ, A + B).
Theorem 2.9. Let the following conditions be satisfied:
1. A ∈ M L(X) and, for some η ∈ (0, 1],
R(λ, A)≤C|λ|
−η
, for sufficiently large λ ∈ Γ,
where Γ is an unbounded set of the complex plane;
2. B is a single valued linear operator in X, D(B) ⊃ D(A),and,foranyε>0,
there exists C(ε) > 0 such that
Bu≤εAu
η
u
1−η
+ C(ε)u, ∀u ∈ D(A);
3. There exists a Banach space Z, D(A) ⊂ Z ⊂ D(B), such that, for some
θ ∈ (0, 1],
R(λ, A)
L(X,Z)
≤ C|λ|
−θ
, for sufficiently large λ ∈ Γ.
Then, every sufficiently large λ ∈ Γ belongs to ρ(A + B) and
R(λ, A + B)≤C|λ|
−η
, for sufficiently large λ ∈ Γ.
Proof. From conditions 1 and 2 we get conditions 1 and 2 of Theorem 2.7. There-
fore, the result of Theorem 2.7 is true. Thus, in order to get the assertion of
Theorem 2.9, it is now enough to show that the operator (λI −A −B)
−1
is single
valued.
Since B is single valued then, for sufficiently large λ ∈ Γ, R(λ, A)B is also
single valued. On the other hand, by condition 3,
R(λ, A)B
L(Z)
≤R(λ, A)
L(X,Z)
B
L(Z,X)
≤ C|λ|
−θ
< 1,
for sufficiently large λ ∈ Γ. Therefore, I − R(λ, A)B has a bounded single valued
inverse operator (I − R(λ, A)B)
−1
in L(Z). Then, since the image R((λI − A −
B)
−1
)=D(A) and the image R(R(λ, A)) = D(A), we obtain, by (2.4), that
the operator (λI − A − B)
−1
is single valued (in fact, (λI − A − B)
−1
=(I −
R(λ, A)B)
−1
R(λ, A), for sufficiently large λ ∈ Γ).
Perturbation Results for Multivalued Linear Operators 117
Remark 2.10. The same conclusion, as in Theorem 2.9, holds if instead of the
inequality in condition 2 it is assumed that, for some σ>0,
BR(λ, A)≤C|λ|
−σ
, for sufficiently large λ ∈ Γ. (2.5)
The proof is the same as that of Theorem 2.9, where the inequality in condition
2 is needed in order to apply Theorem 2.7. In turn, the inequality in Theorem 2.7
has been only used for the proving that BR(λ, A)≤q<1, for sufficiently large
λ ∈ Γ. The latter inequality is now obvious due to (2.5).
3. Modified resolvent approach
Definition 3.1. Let M and L be two single-valued, closed linear operators in a
Banach space X, D(L) ⊂ D(M), 0 ∈ ρ(L). The set {λ ∈ C : λM −L has a single-
valued and bounded inverse defined on X} is called the M modified resolvent set
of L (or simply the M resolvent set of L) and is denoted by ρ
M
(L). The bounded
operator (λM − L)
−1
is called the M modified resolvent of L (or simply the M
resolvent of L).
Theorem 3.2. Let the following conditions be satisfied:
1. Operators M (t) and L(t) are single valued, closed linear operators in a Ba-
nach space X which depend on a parameter t, D(L(t)) ⊂ D(M(t)), every
λ ∈ Γ, |λ|→∞, belongs to ρ
M(t)
(L(t)),and,forsomeη ∈ (0, 1],
M(t)(λM (t) − L(t))
−1
≤C|λ|
−η
, for sufficiently large λ ∈ Γ,
uniformly on t,whereΓ is an unbounded set of the complex plane;
2. An operator L
1
(t) is a single-valued, closed linear operator in X, D(L(t)) ⊂
D(L
1
(t)) and, for any ε>0,
L
1
(t)u≤εL(t)u
η
M (t)u
1−η
+ C(ε)M(t)u, ∀u ∈ D(L(t)),
uniformly on t.
Then, every sufficiently large λ ∈ Γ belongs to ρ
M(t)
(L(t)+L
1
(t)) and
M (t)[λM(t) − (L(t)+L
1
(t))]
−1
≤C|λ|
−η
, for sufficiently large λ ∈ Γ,
uniformly on t.
Proof. Since, for λ ∈ ρ
M(t)
(L(t)),
L(t)(λM(t) − L(t))
−1
=(−(λM(t) − L(t)) + λM(t))(λM(t) − L(t))
−1
= −I + λM(t)(λM(t) − L(t))
−1
then, by condition 1, for sufficiently large λ ∈ Γ, we have
L(t)(λM (t) − L(t))
−1
≤C|λ|
1−η
,