In this section we will present the Carathéodory theorem which will be used to ensure the existence of solution to a Cauchy problem in the interval [0, tm] for every m∈N.
We consider the following Cauchy problem
dY
dt =f(t, Y(t)), t > t0, Y(t0) = Y0.
(A.1)
In the case thatf is a mensurable function we ensure that there is a solution to (A.1) through of the Carathéodory theorem.
Definition A.1 We say that the function f : [0, T]×Ω→Rn satisfies the conditions of Carathéodory onQ= [0, T]×Ω if:
(i) f(t, x) is mensurable in t for each x fixed;
(ii) f(t, x) is continuous in x for each t fixed;
(iii) For eachK ⊂Ωcompact set, there exists an integrable real functionmK(t), such that
kf(t, x)kRn 6mK(t), for all (t, x)∈K.
Theorem A.2 (Caratheodory Theorem)Suppose thatf : [0, T]×Ω→Rnsatisfies the conditions of Carathéodory on Q= [0, T]×Ω. Then there exists a solutionY(t) of (A.1) on some interval |t−t0|6β, where β is a positive constant.
Proof. See Coddington-Levinson [16]
Theorem A.3 (Prolongation Theorem) Let Ω = [0, T] × B with T > 0 and B = {x ∈ Rn;kxkRn 6 b}, where b is a positive constant and k · kRn the norm eu-clidian of the Rn. Suppose that f is a function that satisfies (i), (ii) and that there exists a function m∈L1(0, T) such that
|f(t, x)|6m(t), for all (t, x)∈Ω.
LetY(t)a solution of (A.1)and suppose thatY(t)is defined inI, satisfying|Y(t)|6M with M independent of I and M < b for all t ∈ I. Then Y(t) can to be prolonged in all interval [0, T].
Proof. See Coddington-Levinson [16]
Now we present inequalities frequently used in our work.
Lemma A.4 (Gronwall’s inequality, integral form)Let X ∈L1(0, T;R+)satisfy X(t)6a(t) +
Z t 0
b(s)X(s)ds, a.e. t ∈(0, T) where a, b∈L∞(0, T) and a(·) is increasing. Then,
X(t)6a(t)eR0tb(s)ds, for all t ∈[0,∞].
Proof. See Carvalho, Langa and Robinson [14, Lemma 6.23, p. 167].
Lemma A.5 (Gronwall’s inequality, differential form)LetJ(·)be a non-negative, absolutely continuous function on [0, T], which satisfies for a.e. t the differential in-equality
J0(t)6α(t)J(t) +β(t),
where α(t), β(t) are non-negative, integrable functions in [0, T] . Then, J(t)6J(0)eR0tα(s)ds+
Z t 0
β(τ)eRτtα(s)dsdτ, for all t∈[0, T].
Proof. See Evans [18, Appendix B, p. 624].
Lemma A.6 (Young’s inequality) Let 1< p, q <∞ with 1 p +1
q = 1. Then, ab6 ap
p + bq
q, ∀ a, b>0.
Proof. See Evans [18, Appendix B, p. 622].
Theorem A.7 (Hölder’s inequality) Let Ω ⊂ Rn be a bounded open and 1 6 p, q < ∞ with 1
p + 1
q = 1. If u ∈ Lp(Ω) and v ∈ Lq(Ω) then uv ∈ L1(Ω) and
kuvkL1(Ω) 6kukLp(Ω)kvkLq(Ω).
Proof. See Evans [18, Appendix B, p. 622].
Theorem A.8 (Minkowski’s inequality) If u, v ∈ Lp(Ω) with 1 6 p < ∞ then u+v ∈Lp(Ω) and
ku+vkLp(Ω) 6kukLp(Ω)+kvkLp(Ω).
Proof. See Evans [18, Appendix B, p. 623].
Theorem A.9 (Integration by parts formula) Let u, v ∈C1(Ω). Then Z
Ω
uxivdx+ Z
Ω
uvxi = Z
∂Ω
uvνidS, (i= 1, . . . , n).
Proof. See Evans [18, Appendix C, p. 628].
Appendix B
Linear semigroups
In this chapter we recall a few from theory semigroup of bounded linear operators but with the main objective of presenting the theory of strongly continuous semigroups and analytic semigroups. We present definitions and results of this theory that we use throughout this work. The proof of the results we do not make here, for more details we recommender Carvalho, Langa and Robinson [14], Henry [20] and Pazy [38].
B.1 Definitions and basic concepts
In what follows let X and Y be Banach space over a field K (K= R orK =C) and we denote byL(X, Y)the space of bounded linear operators fromX intoY with the usual norm, that is, for T ∈L(X, Y),
kTkL(X,Y)= sup
x∈X, x6=0
kT xkY kxkX
.
If X = Y we write L(X) to denote L(X, Y). Let X0 be the topological dual of X, that is,X0 =L(X,K)with the norm defined above.
Definition B.1 Asemigroup strongly continuous(or aC0-semigroup) of bounded linear operators is a family of maps {S(t) :t>0} ⊂L(X) such that
(i) S(0) =IX;
(ii) S(t+s) = S(t)S(s), for any t, s>0;
(iii) limt→0+kS(t)x−xkX = 0 or (limt→0+S(t)x=x) for all x∈X.
In general in the space of operators the composition of operators does not commute, however if{S(t) :t>0} ⊂L(X) is a semigroup we have
S(t)S(s) = S(s)S(t), for all t, s>0.
The study of semigroups of linear operators is associated with the study of linear Cauchy problems of the form
du(t)
dt +Au(t) = 0, t >0, u(0) =u0.
(B.1)
where−A:D(A)⊂X →X is linear operator (in general unbounded). The semigroup {S(t) :t >0}is the solution operator associated to (B.1); that is, for each u0 ∈X, the function[0,∞)3t7→S(t)u0 ∈X is the solution (in some sense) of (B.1).
On the other hand given any semigroup of linear operators we can associate it to a differential equation through the following definition.
Definition B.2 Let {S(t) : t > 0} ⊂ L(X) be a C0-semigroup its infinitesimal gen-erator is the linear opgen-erator defined by A:D(A)⊂X →X, where
D(A) =
x∈X : lim
t→0+
S(t)x−x
t exists
and
Ax= lim
t→0+
S(t)x−x
t , for all x∈D(A).
The next result show that all C0-semigroup of bounded linear operator has an exponential bound.
Theorem B.3 Let {S(t) :t>0} ⊂L(X) be a C0-semigroup. There exists constants M >1 and β ∈R such that
kS(t)kL(X) 6M eβt, ∀t >0.
Proof. See Pazy [38, Theorem 2.2, p. 4].
In above theorem if β < 0 we tell that the semigroup has decay exponential or is exponentially stable. If β = 0, that is, kS(t)kL(X) 6M the semigroup is uniformly bounded, moreover if M = 1 it is called aC0-semigroup of contractions.
Now we present some properties of the strongly continuous semigroup which will be the main point in the applications in this work.
Theorem B.4 Let{S(t) :t>0}be a C0-semigroup and A its infinitesimal generator.
The following statements are holds.
(i) For x∈X
[0,∞)3t7→S(t)x∈X is a continuous map;
(ii) The map
[0,∞)3t7→ kS(t)kL(X)
is lower semicontinuous, and therefore mensurable;
(iii) The operator A is closed and densely defined. For each x∈D(A), S(t)x∈D(A) for all t>0, the map
(0,∞)3t7→S(t)x∈X is continuously differentiable and
d+
dtS(t)x=AS(t)x=S(t)Ax, ∀t >0;
(iv) We have that
∞
\
m=1
D(Am) is dense subspace of X;
(v) (Representation of the resolvent operators of A through of Laplace transform of the semigroup) If λ∈Cis such that Re λ > β, where β is given by Theorem B.3, then λ ∈ρ(A) and
(λ−A)−1x= Z ∞
0
e−λtS(t)xdt, for all x∈X.
Proof. See Pazy [38, Theorem 2.4, Corollary 2.5 and Theorem 2.7].
Theorem B.5 Let {S(t) : t > 0} ⊂ L(X) and {T(t) : t > 0} ⊂ L(X) be a C0-semigroup with infinitesimal generator A and B respectively. If A = B then S(t) =T(t), t>0.
Proof. See Pazy [38, Theorem 2.6, p. 6].
We define the resolvent set of a closed linear operator A:D(A)⊂X →X as ρ(A) = {λ ∈C:λ−A is continuous, injective and surjective}.
The setσ(A) =C\ρ(A) is calledspectral set or spectrumof A.
It is easy see by closed graph theorem that, if λ−A is continuous injective and surjective then (λ−A)−1 ⊂ L(X), which is called resolvent operator associated with A.
Remark B.6 The resolvent set ρ(A) is an open set; that is, the spectrum σ(A) = C\ρ(A) is a closed set.
Remark B.7 We consider the Cauchy problem (B.1) such that it is known that −A is the infinitesimal generator of a C0-semigroup {S(t) : t > 0} ⊂ L(X), a direct consequence of Theorem B.4 is the fact that u: [0,∞)→X given by
u(t, u0) =S(t)u0, t >0 is a unique solution of (B.1) (in some sense) such that
u(·, u0) =S(·)u0 ∈C([0,∞);X)∩C1([0,∞);D(A)).
Now we will dedicate the characterization of the infinitesimal generator of a C0 -semigroup. We can characterize an infinitesimal generator of aC0-semigroup through the theorems of Hille-Yosida and Lumer-Phillips.
Theorem B.8 (Hille-Yosida) Let A : D(A) ⊂ X → X a linear operator. Then the following statements are equivalent
(i) A is the infinitesimal generator of a C0-semigroup {S(t) : t >0} ⊂ L(X) such that
kS(t)kL(X)6eβt, for all t >0.
(ii) A is a closed, densely defined linear operator such that ρ(A)⊃(β,∞) and k(λ−A)−1kL(X) 6 1
λ−β, for all λ > β.
Proof. See Pazy [38, Theorem 3.1 and Corollary 3.8, p. 8 and p.12].
LetX∗ be the dual space of the Banach spaceX. We denote the value of x∗ ∈X∗ atx∈X byhx∗, xi orhx, x∗i.
Definition B.9 For every x∈X we define the map duality J :X →2X∗ by J(x) = {x∗ ∈X∗ :Rehx, x∗i=kxk2X, kx∗kX∗ =kxkX}.
Definition B.10 A linear operator A : D(A) ⊂ X → X is dissipative if for every x∈D(A) there exists x∗ ∈J(x) tal que RehAx, x∗i60.
The following result give a characterization of dissipative operators.
Theorem B.11 A linear operator A is dissipative if and only if k(λ−A)xk>λkxk ∀x∈D(A) and λ >0.
Proof. See Pazy [38, Theorem 4.2, p. 14].
Theorem B.12 (Lumer-Phillips) Let A:D(A)⊂X →X be a densely defined linear operator. Then
(i) If A is the infinitesimal generator of a C0-semigroup of contractions on X then A is dissipative and R(λ−A) = X for all λ >0.
(ii) IfAis dissipative andR(λ0−A) = X for someλ0 >0, thenA is the infinitesimal generator of a C0-semigroup of contractions on X.
Proof. See Pazy [38, Theorem 4.3, p. 14].
A direct consequence of the above theorem, and that is used in the applications is given by corollary below.
Corollary B.13 Let A be a linear operator with dense domain D(A) in a Hilbert space H. If A is dissipative and 0 ∈ ρ(A), then A is the infinitesimal generator of a C0-semigoup of contractions on H.
Proof. See Liu and Zheng [27, Theorem 1.2.4, p. 3].
Definition B.14 Let A :D(A) ⊂X →X be a linear operator with D(A) = X. The operator A∗ :D(A∗)⊂X∗ →X∗ defined by
D(A∗) ={x∗ ∈X∗ :∃ y∗ ∈X∗ with hx∗, Axi=hy∗, xi, ∀x∈D(A)}
and
A∗x∗ =y∗, ∀x∗ ∈D(A∗), is called the adjoint operator of A.
The factD(A) =X ensures that there is unique y∗ ∈ X∗ with the property above for some x∗ ∈X∗, that is, D(A∗)6=∅.
Remark B.15 When X is Hilbert space and we identified its topological dual X∗ we have the following
(i) If hAx, yi= hx, Ayi, for all x, y ∈ D(A) holds, we tell that A is symmetric and we denote by A⊂A∗;
(ii) If A=A∗ we tell that A is self-adjoint;
(iii) If A=−A∗ we tell that A is skew-adjoint.
Corollary B.16 Let A be a closed and densely defined linear operator. If both A and A∗ are dissipative, then A is the infinitesimal generator of a C0-semigroup of contractions onX.
Proof. See Pazy [38, Corollary 4.4, p. 15].