Parabolic Systems Involving Sectorial Operators: Existence and Uniqueness of Global Solutions.

In this paper, we study the global existence and uniqueness of sectorial solutions to the system

∂tui+Aiui = fi(u), ∀t∈(0,∞)

ui(0,x) = u0i(x),

whereAiare sectorial operators in the Banach space(X,k·k),

u0i ∈Xandfi:[0,∞)×Xm →Xa given valued function for alli ∈ J1,mK :={1, ...,m}, whereX

m_{is the Banach product} space doted with the normkuk_m=∑m_i=1kuik.

In the casem = 1, since work (Byszewski and Lakshmikant-ham, 1990), (Byszewski, 1991), (Byszewski, 1993), there has been increasing interest in studying abstract problems in Ba-nach Spaces (cf., e.g., (Aizicovici and Mckibben, 2000) and re-ferences therein). For material intimately related to the present paper, we refer to (Henry, 1981), where is studied the existence of sectorial solutions for the single equations. Also in (Jack-son, 1993), (Liang et al, 2002), nonlocal autonomous parabolic

0≤t≤Tis a family ofm-accretive opera-tors inXgenerating a compact evolution family, and the exis-tence of integral solutions to the associated nonlocal problem is shown. In the casem>1, we refer to (Yangari, 2015) in which is studied the existence and uniqueness of mild solutions of a reaction diffusion system with infinitesimal generators. In order to improve the notation, we consider the system

∂tu+Au = f(t,u), ∀t∈(0,∞)

u(0) = u0,

whereu = (ui)mi=1, f = (fi)mi=1andA= diag(A1, ...,Am). Moreover, we consider the normkuk_α = _∑m_i=1kuikα,i on the

spaceXα_=∏m

i Xαi, where the spaceXiαis defined in the next section.

α _{and f}

i

Parabolic Systems Involving Sectorial Operators: Existence

andUniqueness of Global Solutions

Sistemas Parabólicos que Involucran Operadores Sectoriales:

Existencia y Unicidad de Soluciones Globales

Keywords:Reaction diffusion systems, sectorial operators, infinitesimal generators.

[email protected] Received:12/06/2015 Accepted: 23/05/2016 Published: 30/09/2016

Palabras clave:Sistemas de reacción difusión, operadores sectoriales, operadores infinitesimales.

1. INTRODUCTION

1

Miguel Yangari

;

Diego Salazar

Escuela Politécnica Nacional, Departamento de Matemática, Ladrón de Guevara E11-253, Quito, Ecuador

1 1

Abstract: The aim of this paper is to study the existence and uniqueness of global solutions in time to systems ofequations, whenthe diffusion terms are given by sectorial generators.

Resumen: El objetivo de este artículo es estudiar la existencia y unicidad de soluciones globales en tiempo

parasistemas deecuaciones, cuando los términos de difusión están dados por operadores sectoriales.

(1)

Throughout this paper, we assume Ω some open set in R× X

precisely, if(t1,x1)∈Ω, there exists a neighborhoodV⊂Ω such that for(t,x)∈V,(s,y)∈V

kfi(t,x)− fi(s,y)k ≤Li(|t−s|θi+kx−yk_α

for some constants Li > 0and without loss of generality we assumeθ:=θi >0for alli∈J1,mK. Moreover, a solution of

0,t1

0,t1)−→Xmsuch thatu(t0) =u0and on(t0,t1)we ha-ve(t,u(t)) ∈ Ω,u(t) ∈ D(A),∂u

∂t(t)exists,t7→ f(t,u(t)) is locally Hï¿œlder continuous and

Z t₀+ρ

t0

kf(t,u(t))k_mdt<∞

Takingi ∈ _J1,mKfixed throughout this section, we call a li-near operator Ai in a Banach space X, a sectorial operator if it is a closed densely defined operator such that, for some φ∈(0,π/2),M≥1and a realε, the sector

Sε,φ={λ|φ≤ |arg(λ−ε)| ≤π, λ6=ε}

is in the resolvent set ofAiand

(λI−Ai) −1

≤M/|λ−ε| for all λ∈Sε,φ.

Let us note that every bounded linear operator on a Banach space is sectorial. Also, if we define

e−Ait₌ 1 2iπ

Z

Γ(λI+Ai)

−1_eλt_dλ

whereΓis a contour in the resolvent of−Ai witharg(λ) → ±θ as |λ| → ∞ for some θ ∈ (π/2,π), we have that −Ai is the infinitesimal generator of the analytic semigroup (e−Ai(t)_{)t≥0, moreover, ifReσ(A}

i)>bi, then fort>0

e

−Ait ≤ce

−bit_, Aie

−Ait

≤

c te

−bit

It is important to note that, ifB is a bounded linear operator, thene−Btas defined above extends to a group of linear opera-tors and verifies

e−Bte−Bs=e−B(t+s), f or−∞<t,s<∞.

In order to define the fractional power of a sectorial operator

Ai, we assumeReσ(Ai)>0, so, for anyα∈(0, 1)

A−α

i = 1 Γ(α)

Z ∞

0 t

α−1_e−Ait_dt.

Taking A−α

i defined as above, we have that this operator is a bounded linear operator on X which is one-one and satisfies

A−α_i A−β_i = A−(α+β)_i . Furthermore,Aα

i represents the inverse operator ofA−α

i withD(Aαi) = R(A

−α

i )andA0i is the iden-tity on X. An important result concerning positive powers of sectorial operators is

A

α

ie

−Ait ≤cαt

−α_e−bit

withReσ(Ai)>bi>0and ifu∈ D(Aα_i)

(e

−Ait_−I)u

≤

1 αc1−αt

α_kAα

i

also, Aα

iA

β

i = A

β

iAiα = A

α+β

i on D(A

γ

i) with γ = max(α,β,α+β).

Now, we consider the fractional powers ofBi:=Ai+aiIwith

ai ∈Rchosen soReσ(Bi) >0, whereσ(Bi)is the spectrum ofBi. We define the Banach spaceXαi =D(Bαi)with the norm kuk_α,i =

B_iαu

, whereD(Bα_i)is the domain of the operator Bα

i. Finally, takingα≥β≥0, thenXiαis a dense subspace of

X_iβwith continuous inclusion, also,X0_i =X.

For more information about sectorial operators we refer the reader to (Henry, 1981).

In order to state our first result, since−Ai

−A_i(t)₎

t≥0

given by

u(t) =P(t−t0)u0+

Z t

t0

with

P(t) =diag(e−A1(t)_{, ...,e}−Am(t)_).

In what follows, the constantC>0represents different cons-tants.

0,t1

0,t1)intoXα,

Z t₀+ρ

t0

kf(s,u(s))k_mds<∞

0<t<t1 0,t1

3. MAIN RESULTS

the initial value system (1) on(t )is a continuous function

u:[t

for someρ>0and the differential equation (1) is verified.

is the infinitesimal generator of the analytic semigroup(e

for eachi ∈

J1,mK, we define the weak formulation for the system (1)

).

, thenuis a solution of the system(1)on(t

) (2)

. (3)

f or t>0 (4)

uk (5)

P(t−s)f(s,u(s))ds (6)

Lemma 3.1 Ifuis the solution of the system(1)on(t ),

then equation (6) is satisfied. Inversely, if u is a continuous function of(t

0,t1), taking i ∈ J1,mK fixed, we define the auxiliary fun-ction

gi(t,v) = fi(t,u1, ...,ui−1,v,ui+1, ...,un).

Let see thatgi(t,ui(t))is locally Hï¿œlder continuous intand Rt₀+ρ

t0 kgi(t,ui(t))kdt < +∞. Indeed, sincegi : (t0,t1)× Xα

i −→Xand

kgi(t,ui(t))−gi(s,ui(t))k

= kfi(t,u(t))−fi(s,u(s))k ≤ kf(t,u(t))− f(s,u(s))k_m ≤ L|t−s|ν

sincet −→ f(t,u(t))is Hï¿œlder continuous with exponent ν∈(0, 1). Furthermore,

Z t₀+ρ

t0

kgi(t,ui(t)kXdt =

Z t₀+ρ

t0

kfi(t,u(t))kXdt

≤

Z t₀+ρ

t0

kf(t,u(t))k_mdt

< +∞

that

∂tui+Aiui = gi(t,ui),

ui(0) = u0i.

Therefore by the theorem 3.2.2 in (Henry, 1981), we have that

ui

ui(t) =e−Ai(t−t0)u0i+

Z t

t0

e−Ai(t−s)_g

i(s,ui(s))ds.

Repeating the same procedure for alli∈J1,mK, we have

u(t) =P(t−t0)u0+

Z t

t0

P(t−s)f(s,u(s))ds

andu∈ C (t0,t1);Xα. Besides, for eachi∈J1,mK, we have thatui:(t0,t1)−→Xiαis continuous and verifies

ui(t) =e−Ai(t−t0)u0i+

Z t

t0

e−Ai(t−s)_g

i(s,ui(s))ds.

First, we will prove thatuiis locally Hï¿œlder continuous from (t0,t1)toXαi. Thus, ift,t+h∈[t∗0,t∗1]⊂(t0,t1)withh>0 andδi∈(0, 1−α), we claim that

kui(t+h)−ui(t)kα,i≤Cih δ

for some positive constantCi. Indeed,

ui(t+h)−ui(t) = (e−Aih−I)e−Ai(t−t0)u0i

+

Z t

t0

(e−Aih_−I)e−Ai(t−s)_g

i(s,ui(s))ds

+

Z t+h

t e

−A_i(t+h−s)_g

i(s,ui

Now, for anyz∈X, by Theorem 1.4.3 in (Henry, 1981),

(e

−A_ih_−I)e−A_i(t−s)_z

α,i

≤C(t−s)−(α+δi)_hδi_ebi(t−s)_kzk.

Moreover, due to each fi is locally Hï¿œlder int and locally Lipschitz inu, we have that

kfi(t,u(t))− fi(t0,u(t0))k ≤Li(|t−t0|θ+ku(t)−u(t0)k_α)

or equivalently

kgi(t,ui(t))−gi(t0,ui(t0))k ≤ Li(|t−t0|θ+ku(t)kα

+ku(t0)k_α

0,t1)−→ Xαis conti-nuous, then, we have

Cα=ku(t0)kα+_t∗max 0≤t≤t∗1

kuk_α <+∞.

i(t,ui(t))k ≤ L(|t−t0|θ+2Cα) +kgi(t0,u0i)k ≤ L(|t−t0|θ+c).

(e

−Aih_−I)e−Ai(t−t0)_u0i α,i

≤ C(t∗₀−t0)−(α+δi)hδiebi(t1−t0)ku0ik ≤ Chδi_,

sincet∈[t∗₀,t∗₁]⊂(t0,t1

Z t

t0

(e−Aih_−I)e−Ai(t−s)_g

i(s,ui(s))ds

α,i

≤

Z t

t0

C(t−s)−(α+δi)_hδi_ebi(t−s)_kg

i(s,ui(s))kds

≤ Chδi

Z t

t0

(t−s)−(α+δi)_ebi(t−s)_(L|t−t

0|θ+c)ds

≤ Chδi

Z t

t0

(t−s)−(α+δi)_ds Proof.Let assume thatuis the solution of the system (1) on

(t

for someρ>0. Now, sinceuverifies the system (1), we have

namelyusatisfy the equation (6).

Reciprocally, we suppose now thatusatisfy the equation (6)

(7)

is the unique solution of the system (7), which can be writ-ten as

i (8)

(s))ds. (9)

We begin bounding the first term of the equation (9),thus

). Now, let us bound the second term of the equation (9)

). (10) But for hypothesis, we know thatu:(t

Hence, by the inequality (10)

≤ Chδi_.

i) > γi > 0andRe(σ(−aiI)) ≥ −(ai+γi

Rt+h

t e−Ai(t+h−s)gi(s,ui(s))ds

α,i

=

Z t+h

t

B

α

ie−Bi(t+h−s)

e

aiI(t+h−s)_g

i(s,ui(s))

ds ≤

Z t+h

t Cα

(t+h−s)−α_eai(t+h−s)_kg

i(s,ui(s))kds

≤

Z t+h

t Cα(t+h−s)

−α_eai(t+h−s)_(L|s−t

0|θ+c)ds

≤ c

Z t+h

t

(t+h−s)−αeai(t+h−s)_ds

≤ c

Z t+h

t (t+h−s)

−α_e−γt0_e−γt1_e−γh_ds

≤ c

Z t+h

t (t+h−s)

−α_ds

≤ ch1−α ≤ Chδi

i(t,ui(t)) is locally Hï¿œlder continuous on(t0,t1). Indeed,

kgi(t,ui(t))−gi(s,ui(s))k

= kfi(t,ui(t))− fi(s,ui(s))k ≤ Li(|t−s|θ+ku(t)−u(s)kα)

≤ Li(|t−s|θ+ m

∑

i=1

kui(t)−ui(s)kα,i)

≤ Li(|t−s|θ+ m

∑

i=1

Ci|t−s|δi)

≤ C(|t−s|θ₊

m

∑

i=1

|t−s|δi_).

Also,

Z t₀+ρ

t0

kgi(t,ui(t))k ≤

Z t₀+ρ

t0

kf(t,u(ti))kmdt<+∞.

Then, by Theorem 3.2.2 in (Henry, 1981),uisolves the equa-tion

∂tui+Aiui = gi(t,ui(t)) = fi(t,u(t))

ui(0) = u0i

Theorem 3.1 If fi

0,u0)∈Ωthere existsT=T(t0,u0

0,t0+T)with initial conditionu(t0) =u0.

V={(t,x)|t0≤t≤t0+τ,kx−u0k_α≤δ}

is contained inΩand

kfi(t,x)−fi(t,y)k ≤ Likx−ykα

for any(t,x),(t,y) ∈ V. Moreover, we claim that for alli ∈ J1,mK, there exists a constantMi >0such that

B

α

ie

−Ait ≤Mit

−α_eait

for all t > 0. Indeed, taking Re σ(Bi) > γi > 0 and

Re(σ(−aiI))≥ −(ai+γi

B

α

ie−Ait

=

B

α

ie−Aite−aiIteaiIt

=

B α

ie−BiteaiIt

≤

B α

ie−Bit

e

a_iIt ≤ Cαt−αe−γit

e

−(−a_iI)t ≤ Cαt−α_e−γit_Ce(ai+γi)t ≤ Mit−αeait.

Furthermore, we set

κ= max

[t0,t0+τ]

kfi(t,u0)k, L= max

i∈J1,mK

Li

and

a= max

i∈J1,mK

ai, M= max

i∈J1,mK

Mi.

Hence, we can chooseT∈(0,τ)such that

(e

−Aih_−I)u 0i

α,i

≤ δ

2m, f or0≤h≤T

for alli∈J1,mKand

mM(κ+Lδ)

Z T

0 u

−α_eau_du≤ δ

2. for alli∈J1,mK.

Now, we are in position to state our main result in which we es-tablish the existence and uniqueness of solutions to the system (1).

)> 0 such that the system (1) has an unique solution u on (t

verifies the hypothesis (2) for each i ∈ J1,mK, then for any(t

), using inequalities (3) and (4), we have

), using inequalities (3) and (4) Proof.By the previous lemma is enough to find a solutionuof the equation (6). We chooseδ>0,τ>0such that the set

for some large enough positive constant C. Hence, inequa-lity (8) is satisfied. Moreover, t 7→ g

Bounding now the third term of the equation (9), taking

Reσ(B

(11)

IfSdenote the set of continuous functionsy : [t0,t0+T] → Xα_{such thatky(t)−u}

0kα≤δont0≤t≤t0+T, provided

by the norm

kykT = m

∑

i=1

sup{kyi(t)kα,i, t0≤t≤t0+T}

thenSis a complete metric space since it is the product of com-plete metric spaces with the product norm. So, fory ∈ S, we defineH(y):[t0,t0+T]→Xmgiven by

H(y)(t) =P(t−t0)u0+

Z t

t0

P(t−s)f(s,y(s))ds.

We claim thatH : S → Sis a contraction. Indeed, ifY ∈ S

andt06t6t0+T, we have that

kH(y)(t)−u0k_α

≤ k(P(t−t0)−I)u0kα

+

Z t

t0

kP(t−s)f(s,y(s)k_αds

= m

∑

i=1 (e

−Ai(t−t0)_−I)u 0i

α,i

+ m

∑

i=1

Z t

t0 e

−Ai(t−s)_f

i(s,y(s))

α,ids

≤ α

2 + m

∑

i=1

Z t

t0 (Ai)1

α_e−Ai(t−s)

kfi(s,y(s)kds

≤ δ

2+mM

Z t

t0

(t−s)−αea(t−s)(Lky(s)−u0kα+κ)ds

≤ δ

2+mM(κ+Lδ)

Z τ 0 u

−α_eau_du6 δ.

We prove now thatH(y) : [t0,t0+T] → Xα is continuous. Indeed, without loss of generality, we suppose thatz<t, thus

kH(y)(t)−H(y)(z)k_α

= m

∑

i=1

kHi(y)(t)−Hi(y)(z)kα,i

≤ m

∑

i=1

(e

−A_i(t−t0)_−e−Ai(z−t0)_)u 0i

α,i

+ Z z t0

(e−Ai(t−s)_−e−Ai(z−s)_)f

i(s,y(s))ds

α,i

+ Z t z e

−Ai(t−s)_f

i(s,y(s))ds

α,i

= m

∑

i=1

(I1+I2+I3).

Lete>0, we need to findδ>0such that|t−z|<δimplies kH(y)(t)−H(y)(z)k_α<e. Thus, we bound each term of the last inequality

I1= (e

−Ai(t−t0)_−e−Ai(z−t0)_)u0i α,i

=

B α

ie−Ai(z−t0)(e−Ai(t−z)u0i−u0i)

≤ kBα

ik

e

−Ai(z−t0) e

−Ai(t−z)_u 0i−u0i

≤ kBα

ikcie−bi(z−t0)

e

−Ai(t−z)_u 0i−u0i

≤ Ci

e

−Ai(t−z)_u 0i−u0i

<

e 3m

the last inequality is satisfied if|t−z| <δ1for someδ1 >0. Now, boundingI2

I2=

Rz t0(e

−A_i(t−s)_−e−A_i(z−s)_)f

i(s,y(s))ds

α,i

≤ Z z t0 B α

i(e−Ai(t−s)−e−Ai(z−s))fi(s,y(s))

ds

≤ kBα

ik Z z t0 (e

−A_i(t−z)_−I)e−A_i(z−s)_f

i(s,y(s))

ds

≤ kBα

ikC

Z z t0 A α

ie−Ai(z−s)fi(s,y(s))

ds(t−z)

≤ C(t−z)

Z z

t0

Cα(z−s)−αe−bi(z−s)_ds

≤ e

3m

the last inequality is satisfied if|t−z| <δ2for someδ2 >0. Now, proceeding withI3,

I3=

Rt z e

−Ai(t−s)_fi(s,y(s))ds

α,i

≤ kBα

ik Z t z e

−A_i(t−s)

kfi(s,y(s))kds

≤ C

Z t

z e

−bi(t−s)_ds

≤ e

3m

with|t−z| < δ3 for someδ3 > 0. Therefore, takingδ = m´ın{δ1,δ2,δ3}we conclude thatH(y)∈ C [t0,t0+T];Xα

and thenH(y)∈S.

We now prove thatHis a contraction. Lety,z∈ Sandt0 ≤ t≤t0+τ

kHi(y)(t)−Hi(z)(t)k_α,i

≤ Z t t0 B α

ie−Ai(t−s)

kfi(s,y(s))− fi(s,z(s))kds

≤

Z t

t0

Mi(t−s)−αeai(t−s)Liky(s)−z(s)k_α

= ML

Z t

t0

≤ ML

Z T

0 u

−α_eau_dsky−zkT

therefore, withI = [t0,t0+τ]

sup t∈I

kHi(y)(t)−Hi(z)(t)kα,i≤ ML

Z T

0 u

−α_eau_dsky−zkT

for alli∈_J1,mK. Hence

kH(y)(t)−H(z)(t)kT ≤ mML

Z T

0 u

−α_eau_dsky−zkT

≤ 1

2ky−zk T_.

For the fixed point theorem,Hhas an unique solutionu ∈ S, whereu is a continuous function u : [t0,t0+T] → (xαi)m

isatisfies

kfi(t,u)−fi(t0,u0)k ≤ Li(|t−t0|θ+ku−u0k_α)

thus,

kfi(t,u)k ≤ Li(|t−t0|θ+δ) +kfi(t0,u0)k ≤ L(Tθ_{+δ) +B≤C}

and then, for any fixedρ>0

Z t₀+ρ

t0

kf(s,u(s))k_mds = m

∑

i=1

Z t₀+ρ

t0

kfi(s,u(s))kds

≤ mC2ρ<+∞.

0,t0

As a consequence of the previous theorem, we present a result about the behavior of the solution.

Theorem 3.2 If fi

of f(V)is bounded inXm 0 ,t1)andt1

0 ,t2)ift2>t1, then eithert1= +∞or

else there exists a sequencetn →t1− asn → +∞such that (tn,u(tn)) → ∂Ω. IfΩis unbounded, the point at infinity is included in∂Ω.

Proof.Proceeding by contradiction, we suppose thatt1<+∞, but(t,u(t)is not in a neighborhoodNof∂Ωfort≤ t< t1, we can takeNof the formN=Ω\B, whereBis a closed and bounded set ofΩand(t,u(t))∈Bfor allt≤t<t1. We will prove that there exists x1 ∈ Xα such that (t1,x1) ∈ Bwith u(t)→x1inXαwhent→t−1. Even moreu(t1) =x1, which means that the solution could be extended untilt1.

Indeed, letC = sup{kf(t,u)k_m,(t,u) ∈ B}, soC < +∞

because B is bounded and closed and by hypothesis f(B) is bounded inXm_.

Firstly, we can see that ifα≤β<1andt≤t<t1, then

ku(t)k_β = m

∑

i=1

kui(t)k_xβ i

≤ m

∑

i=1

e

−Ai(t−t0)_u 0i

β,i

+

Z t

t0 e

−A_i(t−s)_f

i(s,u(s))

β,i ds

= m

∑

i=1

B

α

iB

β−α

i e

−A_i(t−t₀)_u 0i

+

Z t

t0 B

β

ie

−Ai(t−s)

kfi(s,u(s))kds

≤ m

∑

i=1 ckBα

ik

B

β−α

i e

−Ai(t−t0) ku0ikα,i

+

Z t

t0 B

β

ie

−A_i(t−s)

kfi(s,u(s))kds

≤ c

m

∑

i=1

Mi(t−t0)−(β−α)eai(t−t0)ku0ikα,i

Z t

t0

Mi(t−s)−βeai(t−s)kfi(s,u(s))kds

≤ c

(t−t0)−(β−α)+

Z t

t0

(t−s)−βds

≤ C.

Thus, ku(t)k_β remains bounded when t → t−₁. Now, we suppose thatt≤τ<t<t1, so

u(t)−u(τ) = (P(t−τ)−I)u(τ) +

Z t

τ

P(t−s)f(s,u(s))ds,

then

ku(t)−u(τ)k_α,i

≤ m

∑

i=1

c (e

−Ai(t−τ)_−I)u i(τ)

β,i

+

Z t

τ e

−Ai(t−s)_f

i(s,u(s))

α,ids

≤ m

∑

i=1

ckBα

ik

B

β−α

i (e

−Ai(t−τ)_−I)u i(τ)

+

Z t

τ B

α

ie−Ai(t−s)fi(s,u(s))

ds

.

Bounding the first term, lete1 > 0 an arbitrary number. Sin-ceD(Aβ−α_i )is dense inX, we takev ∈ D(A_iβ−α)such that kui(τ)−vk<η, thus

+T).

Therefore, for the Lemma 3.1,uis solution of the system (1) on(t

. Thenuis a solution of the system (1)on(t is maximal, i.e., there is no solution of the system(1)on(t

B

β−α

i (e

−Ai(t−τ)_−I)ui(τ)

≤

B β−α

i

(e

−A_i(t−τ)_−I)u

i(τ)

≤ c

(e

−A_i(t−τ)_−I)(u

i(τ)−v) + (e−Ai(t−τ)−I)v

≤ c

e

−A_i(t−τ)_(u

i(τ)−v)

+kui(τ)−vk

+ (e

−Ai(t−τ)_−I)v

≤ c

e−bi(t−τ)_ku

i(τ)−vk+kui(τ)−vk

+ (e

−Ai(t−τ)_−I)v

≤ c

e−bit1_{η+η+ (t−τ)}β−α A

β−α

i v

≤ e1+C(t−τ)β−α.

When the last bound is given whenηis small enough such that

c(e−ai(t−τ)_η+η)<e

1. Now, bounding the second term

Z t

τ B

α

ie−Ai(t−s)fi(s,u(s))

ds

≤ c

Z t

τ B

α

ie−Ai(t−s)

kfi(s,u(s))kds ≤ cMi

Z t

τ

(t−s)−α_eai(t−s)_ds

≤ C(t−s)1−α.

Thus, we have

ku(t)−u(τ)k_α≤C

e1+ (t−τ)β−α+ (t−τ)1−α

.

Now, we considertn → t−₁ and let defineun = u(tn), thus, takinge>0, we have

kun−umk_α ≤ C

e1+ (tn−tm)β−α+ (tn−tm)1−α

< e.

ifn,mare large enough. Therefore,(un)is a Cauchy sequen-ce in the complete spasequen-ce Xα_{, thus, there exists x}

1 ∈ Xα such that un → x1, it means (tn,u(tn)) → (t1,x1) and since (tn,u(tn)) ∈ B with B closed, we can conclude that (t1,x1)∈ B.

Also, sinceu:(t0,t1)→Xmis continuous andtn →t−₁, we have that(tn,u(tn))→(t1,u(t1))∈ B, thus, we can conclude thatu(t1) =x1.

To finish the proof, using the Theorem 3.1 and considering (t1,x1) ∈ B ⊂ Ω, we can find an unique solution v on

(t1,t1+T(t1)) for someT(t1

1) =x1. Hence, taking

z(t) =

u(t) i f t∈[t0,t1] v(t) i f t∈[t1,t1+T(t1))

we note thatzis continuous in[t0,t1+T(t1

0) = x0 on (t0,t1+T(t1))which contradict the maximality oft1

Finally, we state that under some extra conditions the unique solution is global in time.

Theorem 3.3 Let us suppose that Ω = (τ,+∞)×Xα _and

fi

more, there existsk(·)a continuous function on(τ,+∞)that verifies

kf(t,u)k_m≤k(t)(1+kuk_α)

for all(t,u)∈ Ω. Ift0 >τ,u0∈ Xα

0) =u0exists for allt>t0.

Proof.Firstly, we can note that hypothesis of the Theorem 3.2 are satisfied. Proceeding by contradiction, we taket0

0,t1)wheret1 is maximal, so, for the last result exists a sequencetn→t−₁ such thatku(tn)k_α→+∞. Howe-ver, sinceβ < αimpliesXα_i < Xβ_i for alli ∈ J1,mK, taking

t∈(t,t1), by a similar procedure to the previous theorem and sinceK(·)is continuous on(τ,∞), i.e., bounded on[t0,t1], we have

ku(t)k_α ≤ C

(t−t0)−(α−β)

+

Z t

t0

(t−s)−α_kf(s,u(s))k

mds

≤ c

(t−t0)−(α−β)

+

Z t

t0

(t−s)−α_{k(s)(1+ku(s)k} αds

≤ c

(t−t0)−(α−β)

+

Z t

t0

(t−s)−α(1+ku(s)k_αds

≤ c+

Z t

t0

c(t−s)−αku(s)k_αds

for the Bellman-Gronwall theorem, we can conclude that

ku(t)k_α ≤ Ce

Rt

t₀c(t−s)−αds

. ) > 0 of the system (1) with initial conditionv(t

)). So, we conclude that z is a solution of the system (1) with z(t

, the unique solution of the system(1)withu(t

>τand

assume that there exists an unique solution of the system (1) defined in(t

Further-≤ C ∀t∈(t,t1).

Which is a contradiction with the fact thatku(tn)k_α → +∞ whentn →t−₁

Similarly to the problem with a single equation, using the pro-perties and estimations of sectorial operators, we state a general result concerning the existence and uniqueness of solutions to systems of equations, when the diffusion terms are given by sectorial generators, also, assuming additional hypothesis on the forcing term, a result of global existence in time is pre-sented. The computations stated in the paper are based in the application of the Banach Fixed Point Theorem.

4. CONCLUSIONS

REFERENCES

.

3, Grade 5 at Escuela Politécnica Nacional.

Benchohra, M and Ntouyas, S. (2001). Nonlocal Cauchy problems for neutral functional differential and integrodif- ferential inclusions in Banach spaces. J. Math. Anal. Appl. 258, 573-590.

ByszewskiL.andLakshmikanthamV.(1990).Theorem about the existence and uniqueness of a solutions ofa non- local Cauchuy problem in a Banach space. Appl. Anal. 40, 11-19.

Byszewski, L. (1991). Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494-505.

Aizicovici S., and Mckibben M. (2000). Existence results for a class of abstractnonlocalCauchyproblems.Nonlinear Analysis. 39, 649-668.

Fu, X. and Ezzinbi, K. (2003). Existence of solutions for neutral functional differential evolution equations with non- local conditions. Nonlinear Analysis. 54, 215-227.

Henry, D. (1981). Geometric theory of semilinear parabolic equations. Berlin, Germany: Springer-Verlag.

Liang, J., Van Casteren, J., and Xiao, T. (2002). Nonlocal Cauchy problems for semilinear evolution equations. Nonli- near Anal. Ser. A: Theory Methods. 50, 173-189.

Ntouyas, S. and Tsamotas, P. (1997). Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl. 210, 679-687.

Ntouyas, S and Tsamotas, P. (1997). Global existence for semilinear integrodifferential equations with delay and non- local conditions. J. Math. Anal. Appl. 64, 99-105

Yangari, M. (2015). Existence and Uniqueness of Glo-bal Mild Solutions for Nonlocal Cauchy Systems in Banach Spaces. Revista Politécnica. 35(2), 149-152.

Miguel Angel Sosa Yangari. Mathematician graduated from Escuela Politécnica Nacional, Ecuador. Doctor of Engineering Sciences,

mention Mathematical Modeling at the

Universidad de Chile, Chile. Doctor in Applied Mathematics at the University of Toulouse, France. Currently is Associate Professor, Level

Diego Salazar Israel Orellana. Mathematician graduated from Escuela Politécnica Nacional, Ecuador and graduate M.Sc. Pure Mathematics, UCE-EPN-USFQ. He is currently Ocasional Professor II at Escuela Politécnica Nacional.

Byszewski,L.(1993).Existenceanduniquenessofsolutions of semilinear evolutionnonlocalCauchyproblem. Zesz. Nauk. Pol. Rzes. Mat. Fiz. 18, 109-112.

Byszewski,L.andAkca,H.(1998).Existenceofsolutions of a semilinear functional-differentialevolution non-local problem. Nonlinear Analysis. 34, 65-72.

Cabré, X. and Roquejoffre, J. (2013). The influence of fractional diffusioninFisher-KPPequation.Commun.Math. Phys. 320, 679-722.

Jackson, D. (1993). Existence and uniqueness of solutions to semilinear nonlocalparabolicequations.J.Math.Anal. Appl. 172, 256-265.

Lin,Y. and Liu, J.(1996). Semilinear integrodifferential equations