Well-posedness of delay parabolic equations with unbounded operators acting on delay terms

R E S E A R C H

Open Access

Well-posedness of delay parabolic equations

with unbounded operators acting on delay

terms

Allaberen Ashyralyev

1,2

_{and Deniz Agirseven}

3*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Trakya University, Edirne, 22030, Turkey

Full list of author information is available at the end of the article

Abstract

In the present paper, the well-posedness of the initial value problem for the delay differential equationdv_dt(t)+Av(t) =B(t)v(t–

ω) +

f(t),t≥0;v(t) =g(t) (–ω≤t≤0) in an arbitrary Banach spaceEwith the unbounded linear operatorsAandB(t) inEwith dense domainsD(A)⊆D(B(t)) is studied. Two main theorems on well-posedness of this problem in fractional spacesEαare established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.

MSC: 35G15

Keywords: delay parabolic equations; well-posedness; fractional spaces; coercive stability estimates

1 Introduction

The stability of delay ordinary differential and difference equations and delay partial dif-ferential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works (see [–] and the references therein) and in-sight has developed over the last three decades. The theory of stability and coercive stabil-ity of delay partial differential and difference equations with unbounded operators acting on delay terms has received less attention than delay ordinary differential and difference equations (see [–]). It is well known that various initial-boundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form

dv(t)

dt +Av(t) =B(t)v(t–ω) +f(t), t≥,

v(t) =g(t) (–ω≤t≤) ()

in an arbitrary Banach spaceEwith the unbounded linear operatorsAandB(t) inEwith dense domainsD(A)⊆D(B(t)). LetAbe a strongly positive operator,i.e.–Ais the gen-erator of the analytic semigroupexp{–tA}(t≥) of the linear bounded operators with exponentially decreasing norm whent→ ∞. That means the following estimates hold:

exp{–tA}_E→E≤Me–δt, tAexp{–tA}_E→E≤M, t>  ()

for someM> ,δ> . LetB(t) be closed operators.

A function v(t) is called a solution of the problem () if the following conditions are satisfied:

(i) v(t)is continuously differentiable on the interval[–ω,∞). The derivative at the endpointt= –ωis understood as the appropriate unilateral derivative.

(ii) The elementv(t)belongs toD(A)for allt∈[–ω,∞), and the functionAv(t)is continuous on the interval[–ω,∞).

(iii) v(t)satisfies the equation and the initial condition ().

A solutionv(t) of the initial value problem () is said to be coercive stable (well-posed) if

Av(t)_E≤ max –ω≤t≤

Ag(t)_E+ sup ≤s≤t

f(t)_E ()

for everyt, –ω≤t<∞. We are interested in studying the coercive stability of solutions of the initial value problem under the assumption that

B(t)A–_E→E≤ ()

holds for everyt≥. We have not been able to obtain the estimate () in the arbitrary Banach spaceE. Nevertheless, we can establish the analog of estimates () where the space

Eis replaced by the fractional spacesEα( <α< ) under an assumption stronger than ().

The coercive stability estimates in Hölder norms for the solutions of the mixed problem of the delay differential equations of the parabolic type are obtained.

The present paper is organized as follows. Section  is introduction. In Section , two main theorems on well-posedness of the initial value problem () are established. In Sec-tion , the coercive stability estimates in Hölder norms for the soluSec-tions of the initial-boundary value problem for delay parabolic equations are obtained. Finally, Section  is our conclusion.

2 Theorems on well-posedness

The strongly positive operatorAdefines the fractional spacesEα=Eα(E,A) ( <α< )

consisting of allu∈Efor which the following norms are finite:

u Eα=sup

λ>

λ–αAexp{–λA}u_E.

We consider the initial value problem () for delay differential equations of parabolic type in the spaceC(Eα) of all continuous functionsv(t) defined on the segment [,∞) with

val-ues in a Banach spaceEα. First, we consider the problem () whenA–andB(t) commute, i.e.

A–B(t)u=B(t)A–u, u∈D(A). ()

Theorem . Assume that the condition

B(t)A–_E→E≤( –α)

holds for every t≥,where M is the constant from().Then for every t, (n– )ω≤t≤nω,

n= , . . . ,we have the following coercive stability estimate:

v(t)_E

α+Av(t)Eα

≤M(α)

max –ω≤t≤

Ag(t)_E α+

n–

k=

max –(k–)ω≤s≤kω

f(s)_E

α+_(n–)max_ω≤s≤tf(s)Eα

, ()

where M(α)does not depend on g(t)and f(t).Here,we putm_k=ak= when m< .

Proof It is clear that

v(t) =u(t) +w(t), ()

whereu(t) is the solution of the problem

du(t)

dt +Au(t) =B(t)u(t–ω), t≥,

u(t) =g(t) (–ω≤t≤), ()

andw(t) is the solution of the problem

dw(t)

dt +Aw(t) =B(t)w(t–ω) +f(t), t≥,

w(t) =  (–ω≤t≤). ()

First, we consider the problem (). Using the formula

u(t) =exp{–tA}g() + t



exp –(t–s)AB(s)g(s–ω)ds, ()

the semigroup property, condition (), and the estimates (), (), we obtain

λ–αAexp{–λA}Av(t)_E

≤λ–αAexp –(λ+t)AAg()_E +λ–α

t



Aexp

–λ+t–s

 A

E→E

B(s)A–_E→E

×Aexp

–λ+t–s

 A

Ag(s–ω)

E

ds

≤ λ–α

(λ+t)–αAg()Eα+  –α M–α

t



Mλ–α_–α

(λ+t–s)–αds_max_≤s≤ωAg(s–ω)Eα

≤ max

–ω≤t≤Ag(t)Eα

for everyt, ≤t≤ωandλ,λ> . This shows that

Au(t)_E

for everyt, ≤t≤ω. Applying mathematical induction, one can easily show that it is true for everyt. Namely, assume that the inequality

Au(t)_E

α≤_–max_ω≤s≤Ag(s)Eα

is true fort, (n– )ω≤t≤nω,n= , , , . . . for somen. Lettingt=s+nω, we have

du(s+nω)

ds +Au(s+nω) =B(s+nω)u

s+ (n– )ω, ≤s≤ω.

Using the estimate (), we obtain

max

nω≤s≤(n+)ω

Au(s–ω)_E

α≤_–(n–)max_ω≤t≤nωAu(t)Eα

for everyt,nω≤t≤(n+ )ω,n= , , , . . . andλ,λ> . This shows that Au(t)_E

α≤_–maxω≤t≤

Ag(t)_E α

for everyt,nω≤t≤(n+ )ω,n= , , , . . . . Therefore _Au(t)_E

α≤_–maxω≤t≤

_Ag(t)_E

α ()

is true for everyt≥. Applying (), the triangle inequality, condition (), and the estimates () and (), we get

u(t)_E

α≤Au(t)Eα+B(t)A

–

E→EAu(t–ω)Eα

≤

 +  –α

M–α

max –ω≤t≤

_Ag(t)

Eα ()

for everyt≥. Second, we consider the problem (). To prove the theorem it suffices to establish the following stability inequality:

Aw(t)_E α≤

M–α

 –α

_n–

k=

max –(k–)ω≤s≤kω

f(s)_E

α+_(n–)maxω≤s≤t

f(s)_E α

()

for the solution of the problem () for every t, (n– )ω≤t≤nω, n= , . . . . Using the formula

w(t) = t



exp –(t–s)Af(s)ds, ()

the semigroup property, and the definition of the spacesEa, we obtain

λ–αAexp{–λA}Aw(t)_E

≤λ–α

t



≤M–α t



λ–α

(λ+t–s)–αf(s)Eαds

≤M–α

t



λ–α

(λ+t–s)–αds

max ≤s≤tf(s)Eα

≤M–α

 –α max≤s≤t

f(s)_E α

for everyt, ≤t≤ωandλ,λ> . This shows that

Aw(t)_E α≤

M–α

 –α max≤s≤t

f(s)_E

α ()

for every t, ≤t≤ω. Applying mathematical induction, one can easily show that it is true for everyt. Namely, assume that the inequality () is true fort, (n– )ω≤t≤nω,

n= , . . . , for somen. Using the formula

w(t) = exp –(t–nω)Aw(nω) + t

nω

exp –(t–s)AB(s)w(s–ω)ds

+ t

nω

exp –(t–s)Af(s)ds, ()

the semigroup property, the definition of the spacesEa, the estimate (), and condition

(), we obtain

λ–αAexp{–λA}Aw(t)_E

≤λ–αAexp –(λ+t–nω)AAw(nω)_E

+λ–α

t nω

Aexp

–λ+t–s

 A

E→E

B(s)A–_E→E

×Aexp

–λ+t–s

 A

Aw(s–ω)

E

ds

+λ–α

t nω

Aexp –(λ+t–s)Af(s)_Eds

≤ λ–α

(λ+t–nω)–αAw(nω)Eα+λ

–α

( –α) t

nω



(λ+t–s)–αAw(s–ω)Eαds +M–α

t nω

λ–α

(λ+t–s)–αf(s)Eαds

≤

λ–α

(λ+t–nω)–α +λ

–α_{( –α)}

t nω



(λ+t–s)–αds

max (n–)ω≤t≤nω

Aw(t)_E α

≤M–α

 –α (n–)maxω≤t≤nω

_n–

k=

max –(k–)ω≤s≤kω

f(s)_E

α+_(n–)max_ω≤s≤tf(s)Eα

+M –α

 –α nωsup≤s≤t

f(s)_E α=

M–α

 –α

_n

k=

max –(k–)ω≤s≤kω

f(s)_E

for everyt,nω≤t≤(n+ )ω,n= , , , . . . andλ,λ> . This shows that

Aw(t)_E α≤

M–α

 –α

_n

k=

max –(k–)ω≤s≤kω

f(s)_E

α+nmaxω≤s≤tf(s)Eα

()

for everyt,nω≤t≤(n+ )ω,n= , , , . . . . Applying equation (), the triangle inequality, and condition () and estimates () and (), we get

w(t)_E

α≤Aw(t)Eα+B(t)A

–

E→EAw(t–ω)Eα+f(t)Eα

≤

 +M

–α

 –α

n

k=

max –(k–)ω≤s≤kω

f(s)_E

α+_nmaxω≤s≤tf(s)Eα

for everyt≥. This result completes the proof of Theorem ..

Now, we consider the problem () when

A–B(t)x=B(t)A–x, x∈D(A)

for somet≥. Note thatAis a strongly positive operator in a Banach spacesE iff its spectrumσ(A) lies in the interior of the sector of angleϕ,  < ϕ<π, symmetric with respect to the real axis, and if on the edges of this sector,S= [z=ρexp(iϕ) : ≤ρ<∞] andS= [z=ρexp(–iϕ) : ≤ρ<∞] and outside it the resolvent (z–A)–is the subject to the bound

(z–A)–_E→E≤ M

 +|z| ()

for someM> . First of all let us give lemmas from the paper [] that will be needed in the sequel.

Lemma . For any z on the edges of the sector,

S=

z=ρexp(iϕ) : ≤ρ<∞

and

S=

z=ρexp(–iϕ) : ≤ρ<∞

and outside it the estimate

A(z–A)–x_E≤M α

Mα( +M)–α(–α)α

α( –α)( +|z|)α x Eα

holds for any x∈Eα.Here and in the future M and Mare the same constants of the

Lemma . Let for all s≥the operator B(s)A–_–A–_{B(s)with domain which coincide}

with D(A)admit a closure Q=B(s)A–_–A–_{B(s)bounded in E.Then for allτ > the}

fol-lowing estimate holds:

A–Aexp{–τA}B(s) –B(s)Aexp{–τA}x_E

≤e(α+ )MαM+α( + M)( +M)–α(–α)α Q E→E x Eα

τ–α_{π α}_{( –α)} .

Here Q=A–_{(AB(s) –B(s)A)A}–_.

Suppose that

A–_{AB(t) –B(t)AA}–

E→E

≤ π( –α)αε

eM+α_M+α

 ( + M)( +M)–α+α–α



( +α) ()

holds for every t≥.Here and in the futureεis some constant, ≤ε≤.

The application of Lemmas . and . enables us to establish the following fact.

Theorem . Assume that the condition

A–_B(t)

E→E≤

( –α)( –ε)

M–α ()

holds for every t≥.Then for every t≥the coercive stability estimate()holds.

Proof In a similar manner as in the proof of Theorem . we establish estimates for the solution of the problems () and (), separately. First, we consider the problem (). Let ≤t≤ωandλ,λ> . Then using (), we have

λ–αAexp{–λA}Au(t)

=λ–αAexp –(λ+t)AAg()

+λ–α

t

 exp

–λ+t–s

 A

B(s)Aexp

–λ+t–s

 A

Ag(s–ω)ds

+λ–α

t

 exp

–λ+t–s

 A

Aexp

–λ+t–s

 A

B(s) –B(s)A

×exp

–λ+t–s

 A

Ag(s–ω)ds

=I+I+I,

where

I=λ–αAexp –(λ+t)A

Ag(),

I=λ–α

t

 exp

–λ+t–s

 A

B(s)Aexp

–λ+t–s

 A

I=λ–α

t

 exp

–λ+t–s

 A

Aexp

–λ+t–s

 A

B(s) –B(s)A

×exp

–λ+t–s

 A

Ag(s–ω)ds.

Using the estimates (), (), and condition (), we obtain

I E=λ–αAexp –(λ+t)A

Ag()_E

≤ λ–α

(λ+t)–αAg()Eα≤

λ–α

(λ+t)–α_–max_ω≤t≤Ag(t)Eα,

I E≤λ–α

t



Aexp

–λ+t–s

 A

E→E

A–_B(s)

E→E

×Aexp

–λ+t–s

 A

Ag(s–ω)

E

ds

≤ max ≤t≤ω

A–_B(t)

E→E

t



Mλ–α_–α

(λ+t–s)–αds_max_≤s≤ωAg(s–ω)Eα

≤ max

–ω≤t≤Ag(t)Eα

 – λ

–α

(λ+t)–α

( –ε)

for everyt, ≤t≤ωandλ,λ> . Now let us estimateI. By Lemma . and using the estimate (), we obtain

I E≤λ–α

t



Aexp

–λ+t–s

 A

E→E

×A–

Aexp

–λ+t–s

 A

B(s) –B(s)Aexp

–λ+t–s

 A

Ag(s–ω)

E

ds

≤λ–αe( +α)M+α_M+α

 ( + M)( +M)–α(–α)α

×

t



A–_{(AB(s) –B(s)A)A}–

E→E–α Ag(s–ω) Eα (λ+t–s)–α_{π α}_{( –α)} ds

≤ max ≤s≤ω

A–_{AB(s) –B(s)AA}–

E→E

×

t



λ–α_{e( +α)M}+α_M+α

 ( + M)( +M)–α(–α)α–α (λ+t–s)–α_{π α}_{( –α)} ds

× max

–ω≤t≤Ag(t)Eα≤_–max_ω≤t≤Ag(t)Eα

 – λ

–α

(λ+t)–α

ε

for everyt, ≤t≤ωandλ,λ> . Using the triangle inequality, we obtain

λ–αAexp{–λA}Au(t)_E≤ max –ω≤t≤

_Ag(t)_E α

for everyt, ≤t≤ωandλ,λ> . This shows that Au(t)_E

α≤_–maxω≤t≤Ag(t)Eα

it suffices to establish the coercive stability inequality () for the solution of the problem (). Now, we consider the problem (). Exactly in the same manner, using (), the semi-group property, and the definition of the spacesEa, we obtain () for everyt, ≤t≤ω.

Applying mathematical induction, one can easily show that it is true for everyt. Namely, assume that the inequality () is true fort, (n– )ω≤t≤nω,n= , . . . for somen. Using () and the semigroup property, we write

λ–αAexp{–λA}Aw(t)

=λ–αAexp –(λ+t–nω)AAw(nω)

+λ–α

t nω

exp

–λ+t–s

 A

B(s)Aexp

–λ+t–s

 A

Aw(s–ω)ds

+λ–α

t nω

exp

–λ+t–s

 A

Aexp

–λ+t–s

 A

B(s) –B(s)A

×exp

–λ+t–s

 A

Aw(s–ω)ds+λ–α

t nω

Aexp –(λ+t–s)Af(s)ds

=I+I+I+I,

where

I=λ–αAexp –(λ+t–nω)A

Aw(nω),

I=λ–α

t nω

exp

–λ+t–s

 A

B(s)Aexp

–λ+t–s

 A

Aw(s–ω)ds,

I=λ–α

t nω

exp

–λ+t–s

 A

Aexp

–λ+t–s

 A

B(s) –B(s)A

×exp

–λ+t–s

 A

Aw(s–ω)ds,

I=λ–α

t nω

Aexp –(λ+t–s)Af(s)ds.

Using the estimate () and condition (), we obtain

I E=λ–αAexp –(λ+t–nω)A

Aw(nω)_E≤ λ –α

(λ+t–nω)–αAw(nω)Eα,

I E≤λ–α

t nω

Aexp

–λ+t–s

 A

E→E

A–_B(s)

E→E

×Aexp

–λ+t–s

 A

Aw(s–ω)

E

ds

≤ max

nω≤t≤(n+)ω

A–_B(t)

E→E

t nω

Mλ–α_–α

(λ+t–s)–αds_nω≤max_{s≤(n+)ω}Aw(s–ω)Eα

≤

 – λ

–α

(λ+t–nω)–α

( –ε) max (n–)ω≤s≤nω

Aw(s)_E α,

I E≤M–α

t nω

λ–α

(λ+t–s)–αf(s)Eαds≤

M–α

 –α nωsup≤s≤t

for every t, nω≤t≤(n+ )ω, n= , , , . . . , and λ, λ> . Now let us estimate I. By Lemma . and using the estimate () and condition (), we obtain

I E≤λ–α

t nω

Aexp

–λ+t–s

 A

E→E

×A–

Aexp

–λ+t–s

 A

B(s) –B(s)Aexp

–λ+t–s

 A

Aw(s–ω)

E

ds

≤λ–αe( +α)M+αM_+α( + M)( +M)–α(–α)α

×

t nω

A–_{(AB(s) –B(s)A)A}–

E→E–α Aw(s–ω) Eα (λ+t–s)–α_{π α}_{( –α)} ds

≤ max ≤s≤ω

A–_{AB(s) –B(s)AA}–

E→E

×

t nω

λ–αe( +α)M+α_M+α

 ( + M)( +M)–α(–α)α–α (λ+t–s)–α_{π α}_{( –α)} ds

× max

(n–)ω≤s≤nω

Aw(s)_E α

≤

 – λ

–α

(λ+t–nω)–α

ε max (n–)ω≤s≤nω

Aw(s)_E α

for everyt,nω≤t≤(n+ )ω,n= , , , . . . andλ,λ> . Using the triangle inequality and estimates for all Ik E,k= , , , , we obtain

λ–αAexp{–λA}Aw(t)_E≤M –α

 –α

_n

k=

max –(k–)ω≤s≤kω

f(s)_E

α+_nmaxω≤s≤tf(s)Eα

for everyt,nω≤t≤(n+ )ω,n= , , , . . . andλ,λ> . This shows that

Aw(t)_E α≤

M–α

 –α

_n

k=

max –(k–)ω≤s≤kω

f(s)_E

α+_nmaxω≤s≤tf(s)Eα

for everyt,nω≤t≤(n+ )ω,n= , , , . . . . This result completes the proof of

Theo-rem ..

Note that these abstract results are applicable to the study of stability of various

de-lay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study the structure ofEαfor space operators

in Banach spaces. The structure ofEαfor some space differential and difference operators

in Banach spaces has been investigated (see [–]). In Section , applications of The-orem . to the study of the coercive stability of initial-boundary value problem for delay

3 Applications

First, we consider the initial-boundary value problem for one dimensional delay differen-tial equations of parabolic type

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂u(t,x)

∂t –a(x)

∂u(t,x)

∂x +δu(t,x)

=b(t)(–a(x)∂u_∂(t_x–ω,x)+δu(t–ω,x)) +f(t,x),  <t<∞,x∈(,l), u(t,x) =g(t,x), –ω≤t≤,x∈[,l],

u(t, ) =u(t,l) = , –ω≤t<∞,

()

wherea(x),b(t),g(t,x),f(t,x) are given sufficiently smooth functions andδ>  is a suf-ficiently large number. We will assume thata(x)≥a> . The problem () has a unique smooth solution. This allows us to reduce the initial-boundary value problem () to the initial value problem () in Banach spaceE=C[,l] with a differential operatorAxdefined by the formula

Axu= –a(x)d _u

dx +δu ()

with domainD(Ax_{) ={u∈C}()_{[, ] :u() =u() = }. Let us give a number of corollaries} of the abstract Theorem ..

Theorem . Assume that

sup ≤t<∞

b(t)≤  –α

M–α. ()

Then for all t≥the solutions of the initial-boundary value problem()satisfy the fol-lowing coercive stability estimates:

u(t,·)_Cα_[,l]+u(t,·)_C+α_[,l]

≤M(α)

max –ω≤t≤

g(t,·)_C+α_[,l]

+

n

k= max –(k–)ω≤s≤kω

f(s,·)_Cα_[,l]+ max

nω≤s≤tf(s,·)Cα[,l]

,  <α<  ,

where M(α)is not dependent on g(t,x)and f(t,x).Here Cβ_{[,l]is the space of functions} satisfying a Hölder condition with the indicatorβ∈(, ).

The proof of Theorem . is based on the estimate

exp –tAx_{C[,l]→C[,l]}≤M, t≥,

and on the abstract Theorem ., on the strong positivity of the operatorAx_{inC[,l] (see}

[, ]) and on Theorem . on the structure of the fractional spaceEα=Eα(C[,l],Ax)

for  <α<_.

Second, we consider the initial nonlocal boundary value problem for one dimensional delay differential equations of parabolic type,

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂u(t,x)

∂t –a(x)

∂u(t,x)

∂x +δu(t,x)

=b(t)(–a(x)∂u_∂(t_x–ω,x)+δu(t–ω,x)) +f(t,x),  <t<∞,x∈(,l), u(t,x) =g(t,x), –ω≤t≤,x∈[,l],

u(t, ) =u(t,l), ux(t, ) =ux(t,l), –ω≤t<∞,

()

wherea(x),b(t),g(t,x),f(t,x) are given sufficiently smooth functions andδ>  is a suf-ficiently large number. We will assume thata(x)≥a> . The problem () has a unique smooth solution. This allows us to reduce the initial-boundary value problem () to the initial value problem () in Banach spaceE=C[,l] with a differential operatorAx_defined

by the formula

Axu= –a(x)d _u

dx +δu ()

with domainD(Ax_{) ={u∈C}()_{[, ] :u() =u(),u() =u()}. Let us give a number of} corollaries of the abstract Theorem ..

Theorem . Assume that condition()holds.Then for all t≥the solutions of the initial-boundary value problem()satisfy the following coercive stability estimates:

u(t,·)_Cα_[,l]+u(t,·)_C+α_[,l]

≤M(α)

max –ω≤t≤

g(t,·)_C+α_[,l]

+

n

k= max –(k–)ω≤s≤kω

f(s,·)_Cα_[,l]+ max

nω≤s≤tf(s,·)Cα[,l]

,  <α<  ,

where M(α)is not dependent on g(t,x)and f(t,x).

The proof of Theorem . is based on the estimate

exp –tAx_{C[,l]→C[,l]}≤M, t≥,

and on the abstract Theorem ., on the strong positivity of the operatorAx _inC[,l]

(see []) and on Theorem . on the structure of the fractional spaceEα=Eα(C[,l],Ax)

for  <α<_.

Theorem . Forα∈(,_),the norms of the space Eα(C[,l],Ax)and the Hölder space Cα_{[,l]are equivalent[].}

Third, we consider the initial value problem on the range

≤t≤,x= (x, . . . ,xn)∈Rn,r= (r, . . . ,rn)

for mth order multidimensional delay differential equations of parabolic type,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

∂u(t,x)

∂t +

|r|=maτ(x) ∂

|r|_u(t,x)

∂xr_···∂xnrn +δu(t,x) =b(t)(_|r|=maτ(x)∂

|r|_u(t–ω,x)

∂xr···∂xrnn +

δu(t–ω,x)) +f(t,x),  <t<∞,x∈Rn_,

u(t,x) =g(t,x), –ω≤t≤,x∈Rn_,|r|=r

+· · ·+rn,

()

wherear(x),b(t),g(t,x), andf(t,x) are sufficiently smooth functions andδ>  is a

suffi-ciently large number. We will assume that the symbol [ξ= (ξ, . . . ,ξn)∈Rn]

Ax_(ξ) = |r|=m

ar(x)(iξ)r· · ·(iξn)rn

of the differential operator of the form

Ax_= |r|=m

ar(x)

∂|r|

∂xr

 · · ·∂x

rn

n

, ()

acting on functions defined on the spaceRn_{, satisfies the inequalities}

 <M|ξ|m≤(–)mAx(ξ)≤M|ξ|m<∞

forξ= , where|ξ|=|ξ|+· · ·+|ξn|. The problem () has a unique smooth solution.

This allows us to reduce the initial value problem () to the initial value problem () in Banach spaceEwith a strongly positive operatorAx_=Ax

+δIdefined by (). Let us give a number of corollaries of the abstract Theorem ..

Theorem . Assume that condition()holds.Then for all t≥the solutions of the initial-boundary value problem()satisfy the following coercive stability estimates:

u(t,·)_Cmα_(Rn₎+

|r|=m

_∂x∂r|r|u(t,·)

 · · ·∂x

rn

n

Cmα_(Rn₎

≤M(α)

max –ω≤t≤

|r|=m

_∂x∂r|r|g(t,·)

 · · ·∂x

rn

n

Cmα_(Rn₎ +

n

k= max –(k–)ω≤s≤kω

f(s,·)_Cmα_(Rn₎+ max

nω≤s≤tf(s,·)Cmα(Rn)

,  <α<  m,

where M(α)does not depend on g(t,x)and f(t,x).Here Cε(Rn)is the space of functions

satisfying a Hölder condition with the indicatorε∈(, ).

The proof of Theorem . is based on the estimate

exp –tAx_C(Rn_)→C(Rn₎≤M, t≥,

and on the abstract Theorem ., on the strong positivity of the operatorAxinC(Rn), and on the equivalence of the norms in the spacesEα=Eα(A,C(Rn)) andCmα(Rn) when

4 Conclusion

In the present paper, two theorems on the well-posedness of the initial value problem for the delay parabolic differential equations with unbounded operators acting on delay terms in fractional spacesEαare established. In practice, the coercive stability estimates

in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Fatih University, Istanbul, 34500, Turkey.}2_{Department of Mathematics, ITTU, Gerogly}

Street, Ashgabat, 74400, Turkmenistan.3_{Department of Mathematics, Trakya University, Edirne, 22030, Turkey.} Acknowledgements

This work is supported by Trakya University Scientific Research Projects Unit (Project No: 2010-91).

Received: 31 March 2014 Accepted: 8 May 2014 Published:20 May 2014

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