Scholarly article on topic 'Regularity properties of degenerate convolution-elliptic equations'

Regularity properties of degenerate convolution-elliptic equations Academic research paper on "Mathematics"

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Academic research paper on topic "Regularity properties of degenerate convolution-elliptic equations"

Musaev and Shakhmurov Boundary Value Problems (2016) 2016:50 DOI 10.1186/s13661-016-0555-1

0 Boundary Value Problems

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Regularity properties of degenerate convolution-elliptic equations

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Hummet K Musaev1 and Veli B Shakhmurov2

"Correspondence: veli.sahmurov@okan.edu.tr 2Department of Mechanical Engineering, Okan University, Akfirat Beldesi, Tuzla, Istanbul, 34959, Turkey

3Department of Mathematics, Azerbaijan Khazar University, Baku, Azerbaijan

Fulllist of author information is available at the end of the article

Abstract

The coercive properties of degenerate abstract convolution-elliptic equations are investigated. Here we find sufficient conditions that guarantee the separability of these problems in Lp spaces. It is established that the corresponding convolution-elliptic operator is positive and is also a generator of an analytic semigroup. Finally, these results are applied to obtain the maximal regularity properties of the Cauchy problem for a degenerate abstract parabolic equation in mixed Lp norms, boundary value problems for degenerate integro-differential equations, and infinite systems of degenerate elliptic integro-differential equations.

MSC: 34G10; 45J05; 45K05

Keywords: positive operators; abstract weighted spaces; operator-valued multipliers; boundary value problems; convolution equations; integro-differential equations

ft Spri

ringer

1 Introduction, notations, and background

In recent years, maximal regularity properties for differential operator equations, especially of parabolic and elliptic type, have been studied extensively e.g. in [1-14] and the references therein. Moreover, convolution-differential equations (CDEs) have been treated e.g. in [2,15-19] (for comprehensive references see [18]). Convolution operators in Banach-valued spaces were studied e.g. in [8-10, 20-23]. However, the convolution-differential operator equations (CDOEs) are a relatively poorly investigated subject. In [2] the parabolic type CDEs with bounded operator coefficients were investigated. In [9] the regularity properties of degenerate ordinary CDOEs were studied. The main aim of the present paper is to study the following degenerate elliptic CDOEs:

* D[a] u + (A + X) * u = f (x) (1.1)

and the Cauchy problem for the degenerate parabolic CDOE

— + ^ aa *D[a]u + A * u = f(t,x), u(0,x) = 0, 91

in E-valued Lp spaces, where E is a Banach space, A = A(x) is a linear operator in E, aa = aa(x) are complex-valued functions, a = (ai,a2,...,an), X is a complex number, y = y(x)

© 2016 Musaev and Shakhmurov. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tionalLicense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution,and reproduction in any medium, provided you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

is a positive measurable function on n c Rn, and

/ d \a'

DM = D^Df]...DM dK] = Y(x)_ .

XI X2 xn xi y dx /

Here the convolutions aa *D[a] u and A * u are defined in the distribution sense (see e.g. [1]).

In applications, particularly the above equations describe the charged particle motion for certain configurations of oscillating magnetic fields (see e.g. [24]). Maximal regularity has proven very useful in handling some concrete non-linear evolution equations as shown by [25] and [23], which deal with the Navier-Stokes equations of fluid dynamics. One of the main features of the present work is that the convolution equations are degenerate on some points of R =(-rc, <xi). Moreover, equation (1.1) has a variable operator coefficient. Since such a type equations occur in applications, it is important to show the existence and uniqueness of the solution. In this paper, we establish the separability properties of the problem (1.1) and the maximal regularity of Cauchy problem for parabolic CDOE. Moreover, we prove that the operator generated by problem (1.1) is positive. The main tools of this work are the operator-valued Fourier multipliers. Since equation (1.1) has unbounded operator coefficients there occurs some difficulty. This fact is derived by using the representation formula for a solution of the problem (1.1) and operator-valued multiplier in Lp,Y (Rn; E).

Let E be a Banach space and y = Y(x), x = (xi,x2,...,xn) be a positive measurable weighted function on a measurable subset n e Rn. Let LpY (fi; E) denote the space of strongly E-valued functions that are defined on n with the norm

\\Lp,y = lf llLp,y (Q;E) =

-/(¡f (x)IIE Y (x) dxflp, 1 < p < rc.

For y(x) = 1, the space LpY(n,E) will be denoted by Lp = Lp(n;E). If IIl,x,,p,y (n;E) = ess sup [ y (x)| f(x)|| J.

The weight y(x) we will consider to satisfy an Ap condition; i.e., y(x) e Ap, 1 <p < rc,if there is a positive constant C such that

hly(x)dx)(ïQïLy_pil(x)dxl -c

for all compacts Q c Rn. Let C be the set of complex numbers and

Sv = {X;X e C, | argU{0}, 0 < y < n.

Let E1 and E2 be two Banach spaces and let £(E1, E2) denote the spaces of bounded linear operators acting from E1 to E2. ForE1 = E2 = E we denote B(E,E) byB(E).

A closed linear operator function A = A(x) is said to be uniformly y-positive in Banach space E, if D(A(x)) is dense in E and does not depend on x and there is a positive constant M so that

|| (A(x) + XI )-1|| < M(1+|X|)-1

for every x e Rn and X e Sp, p e [0, n), where I is an identity operator in E. Sometimes instead of A + XI we will write A + X and it will be denoted by AX. It is well known [11], Section 1.14.1, that there exist fractional powers Ae of the positive operator A. Let E(Ae) denote the space D(Ae) with the graphical norm

\\u\\E(Ae) = (\\u\\p + \\A9 uf) p, 1 < p < TO,-TO < 0 < TO.

Let S = S(Rn;E) denote a Schwartz class, i.e. the space of E-valued rapidly decreasing smooth functions on Rn equipped with its usual topology generated by seminorms. S(Rn; C) is denoted by just S.

Let S'(Rn;E) denote the space of all continuous linear operators, L : S ^ E, equipped with the bounded convergence topology. Recall S(Rn;E) is norm dense in Lp,Y (Rn;E) when 1 <p < to, y e Ap.

Let ^ be a domain in Rn. C(^,E) and Cm(Q;E) will denote the spaces of E-valued bounded uniformly strongly continuous and m-times continuously differentiable functions on respectively.

Let a = (a1, a2,..., an), where ai are integers. An E-valued generalized function Daf is called a generalized derivative in the sense of Schwartz distributions of the function f e S(Rn, E), if the equality

{Daf, p) = (-1)wf, Dap)

holds for all p e S.

Let F denote the Fourier transform. Through this section the Fourier transformation of a functionf will be denoted byf, Ff = f, and F-1f = f. It is well known that

F (Df = (i^1)a1 • • • fer/, Da (F f)) = F [(-ix1)a1 • • • (-ix„)anf ] for all f e S'(Rn; E).

Suppose E1 and E2 are two Banach spaces. A function ^ e LTO(Rn;B(E1,E2)) is called a multiplier from Lp,Y (Rn; E1) to Lp,Y (Rn; E2) for p e (1, to) if the map u ^ Tu = F-1^ (f )Fu, u e S(Rn;E1) is well defined and extends to a bounded linear operator

T : Lp>y(.Rn;E^ ^ Lp,Y{Rn;E2).

The space of all Fourier multipliers from Lp,Y (Rn;E1) to Lp,Y(Rn;E2) will be denoted by MpY(E1,E2). For E1 = E2 = E we denote MppYY(E1,E2) by M^(E). A Banach space E is called a UMD-space [26, 27] if the Hilbert operator

(Hf)(x) = lim / M- dy

\x-y\>s x -y

is bounded in Lp(R;E),p e (1, to) (see e.g. [4]). UMD spaces include e.g. Lp, lp spaces and the Lorentz spaces Lpq,p, q e (1, to).

A set K c B(E1,E2) is called R-bounded (see [4, 5,15, 28]) if there is a constant C > 0 such that, for all T1, T2,...,Tm e K and u1,u2,...,um e E1, m e N

»1 m .

/ Y]rj(y)TjUj dy - c

Jo )=1 £2 Jo

J2r'(y)i

where {rj} is a sequence of independent symmetric {-1;1}-valued random variables on [0,1] and N denotes the set of natural numbers. The smallest C for which the above estimate holds is called the R-bound of K and is denoted by R(K).

Definition 1.1 A Banach space E is said to be a space satisfying the weighted multiplier condition if for any V e C(n)(Rn\{0}; B(E))the R-boundedness of the set {|f ||P|dPv (f ):f e Rn\{0},P = (Pi,P2,...,Pn),Pk e {0,1}} implies that V is a Fourier multiplier inLPiY(Rn;E), i.e., V e Mpp'YY(E) for anyp e (1, rc).

Remark 1.1 Note that, if E is a UMD space, then by virtue of [5, 6,13, 21] it satisfies the multiplier condition.

Definition 1.2 A positive operator A(x), x e Rn is said to be uniformly R-positive in a Banach space E if there exists a y = yA e [0, n) such that the set LA = {f (A + f )-1: f e Sy} is uniformly R-bounded, i.e.

sup R({[A(x)(A(x) + fI)-1]: f e Sy}) < M.

Note that in Hilbert spaces every norm bounded set is R-bounded. Therefore, in Hilbert spaces all positive operators are R-positive. Let h e R, m e N, and ek, k = 1,2,..., n, be standard unit vectors of Rn. Let

Ak(h)f (x) =f (x + hek) -f (x).

Let A = A(x), x e Rn be closed linear operator in E with domain D(A) independent of x. The Fourier transformation of A(x) is a linear operator with the domain D(A) defined as

Au(y) = Au(y) for u e S'(Rn;E(A)), y e S(Rn).

(For details see [1].) A(x) is differentiable if we have the limit

dA \ Ak(h)A(x)u , , N

— u = lim---, k = 1,2,...n,u e D(A),

dxk) h^0 h

in the sense of the E-norm.

Let E0 and E be two Banach spaces, where E0 is continuously and densely embeds into E. Let l be a integer number. WlpY, (Rn;E0, E) denotes the space of all functions from S'(Rn; E0)

such that u e Lp,Y(Rn;E0) and the generalized derivatives Dlk u = H e Lp,Y (Rn; E) withthe

I|u|Wp,Y (Rn-f0,E) = i|u|ilp,y (Rn;£0) + lDkulip,Y (Rn-E) < rc.

It is clear that

WlptY(Rn;£„,E) = WlptY(Rn;E) nLw (Rn;£0). Consider the problem

* Da u + (A + k) * u = f (x), (1.2)

where A = A(x) is a linear operator in E, aa = aa (x) are complex-valued functions, a = (a1, a2,...,an), k is a complex parameter.

A function u e WlPtY (Rn; E(A), E) satisfying equation (1.2) a.e. on Rn is called the solution of equation (1.2).

The elliptic CDOE (1.2) is said to be uniform separable in Lp,Y (Rn; E) ifequation (1.2) has a unique solution u forf e LPyY(Rn;E)and the following coercive estimate holds:

J2 || aa * Da u | LpY (Rn-E) + llA * u\\Lp,y (Rn& < c\f \\lPy (Rn;E),

where the positive constant C is independent off. In a similar way to [3], Theorem A0, we obtain the following.

Proposition A0 LetE be a UMD space and y e Ap. Assume is a set of operator functions from Cn(Rn\{0}; B(E)) depending on the parameter h e Q e R and there is a positive constant K such that

supR({l$l^D^h^): £ e Rn\{0},pk e {0,1}}) < K.

Then the set is a uniformly bounded collection of Fourier multipliers in Lp>Y (Rn; E). Let Ei and E2 be two Banach spaces. Suppose T e B(E\,E2) and 1 < p < to. T e B(Lp(Rn;E1),Lp(Rn;E2)) will denote the operator (Tf)(x) = T(f(x)) forf e Lp(Rn;E1) and x e Rn.

From [10] we have the following.

Proposition A1 Let 1 < p < to. IfW c B(E1,E2) is R-bounded, then the collection W = {T: T e W}c B(Lp(Rn;E1),Lp(Rn;E2)) is also R-bounded.

2 Elliptic CDOE

Condition 2.1 Assume A(£) is a uniformly positive operator in E and aa e LTO(Rn) such that

L(£) ^ a a (£ )(i£ )a e Sn, \L(£) | > ^ la k ll£k l,

lal<l k=1

k = k(a) = (a1, a2,...,an), ak = l, ai = 0,i = k,

e [0,n), f = (|i,&,..., fn) e Rn,

where ak denote the coefficients of flk in the polynomial L(f ). Consider

ao(f, k) = kD(f, k), ai(f, k) = A (f )D(f, k), |k|i-~Taa (f )(if )a D(f, k),

D(f, k) = [A (f ) +L(f ) + k]-i.

Lemma 2.1 Assume the Condition 2.i holds and k e SV2 with y2 e [0, n), where yA + + y2 < n, then the operator functions ai(f, k) are uniformly bounded, i.e.,

||ai(f,k)||B(£) < C, i = 0,i,2.

Proof By virtue of [4], Lemma 2.3, for L(f ) e Sw, k e SV2, and + y2 < n, there is a positive constant C such that

\k + L(S)| > C(\k\ + \L(Ç))•

Since L(f) e Sw, in view of (2.1) and the resolvent properties of positive operators, we see that A (f) + L(f) + k is invertible and

1+M|k + L(f)|(i+|k + L(f)|) i <Mi. Next, let us consider a2. It is clear that

h(f,k)\m < C£ |k|f[[|f ||X|-1 ]ak|D(f,k)\m.

<l k=1

Since A is uniformly positive and L(f) e Sw, setting yk = (|k| 1 |fk|)ak in the following well-known inequality:

yT y? ---fn < c(l + £ ykj, yk > 0, \a\<l,

we obtain

M, ^lU < C£ w

i + £ \Sk\lw

\k+us r1.

Taking into account the Condition 2.i and (2.i)-(2.3) we get ||a2(f,k)||BE) < c( |k| + ^^ |fk|M (|k| + |L(f)|)-i < C.

Lemma 2.2 Assume the Condition 2.1 holds. Suppose aa e C(n)(Rn), [DPA(£)]A-1(£0) e C(Rn;B(E)) and

l£l lPl \Daa(£) \ < C1, fa e {0,1}, £ e Rn\{0}, 0 <P l < n, || l£lWlDA(£)]A 1 (£0)||B(E) < C2, Pk e {0,1},£,£0 e Rn\{0}.

Then the operator functions l£ l lPlDPai(£, k), i = 0,1,2, are uniformly bounded.

Proof Let us first prove that £k is uniformly bounded. In fact,

< ii/l nb(e) + 11/2 iib(e) + 11^3 nb(e),

d A (H )

D(H, k), I2= A (H )

d A (H )

d2(H, k)

I3= A (H )

D2(H, k).

By using (2.3) and (2.5) we get

d A (H )

iiIiiib(e) <

A -i(Ho)

i|oi||b(e) < C.

Due to the positivity of A, by using (2.3) and (2.5) we obtain

d A (H ) ■

1112 iib(e) <

A-x(Ho)

I0lllB(E) < C.

Since A (£) is uniformly positive, by using (2.1), (2.3), and (2.5) for k e S((p2) and p1 + p2 < n we get

1113 iib(e) <

\\D(H, k)|| b(eJI ai(H,

IIb(e) < c.

In a similar way, the uniform boundedness of a0(£, k) is proved. Next we shall prove £k d2 is uniformly bounded. Similarly,

, d02 Hk 1H2

< ii/inb(e) + II/2 iib(e),

/ = E lk|1-^(Hk dg) [(iH )a + aa (H )m (iH )a ]d(H , k),

/2 = £ | k |V (f )(lf

I a | <l

'fc daa a fc 9A(f)■

+ aa (f )(lf )a + fk-tt-dfk dfk

D2 (f, k).

First of all, we show J is uniformly bounded. Since

y/iy,(£) < ^fk ifa iki1-D(f, k)i ^ dfk

due to positivity of A, by virtue of (2.1) and (2.3)-(2.5) we obtain HAH^e < C. In a similar way we have ||/2 |B(E) < C. Hence, the operator functions £k, i = 0,1,2 are uniformly bounded. From the representations of oi(£, X) it easy to see that operator functions fD^O^, X) contain similar terms to Ik; namely, the functions ll^lD^oi(£, X) will be represented as combinations of principal terms

DV A (f )+ D aa (f )] D(f, k)f, £ |k|1-^ f'D[A(f ) + aa(f)][D(f,k)f,

where //, v are «-dimensional integers vectors and |x| + |v| < \p|. Therefore, by using similar arguments to the above and in view of (2.6) one can easily check that

|£,k)||B(E) < C, i = 0,1,2.

From [10] we obtain the following. □

Lemma 2.3 Let all conditions of Lemma 2.2 hold. Suppose E is a Banach space satisfying the uniform multiplier condition. Then the sets

So(H,k) = {|£ ^Dfoofê,k);£ e Rn\{0}}, Si(f,k)= {|£,k);£ e Rn\{0}}, S2(£,k) = {|£^D^fê,k);£ eRn\{0}}

are uniformly R-boundedfor jik e {0,1} and 0 <|j81< n.

Result 2.1 Suppose all conditions of the Lemma 2.2 are satisfied, E is a UMD space. Then the sets Si(£,k), i = 0,1,2, are uniformly R-bounded.

Now we are ready to present our main results. We find sufficient conditions that guarantee the separability of problem (1.2).

Condition 2.2 Suppose the following are satisfied:

L(f )=£ a a (f )(lf )a e V Vi e [0, n ), |L(f )| > C^ |a k ||fk |r, f e Rn;

|a |<l

(2) a a e C(n)(Rn) and

if âa(f) \ < Ci, e (0,11,0 <\p\<n;

(3) for 0 < \3 \ <n,

[D3A(f)]Â-1(fo) e C(Rn;B(E)), \f \03\\\[D3A(f)]aA-1(fo)||B(E) < C2, f, fo e Rn\(0}.

X = LPy (Rn;E), Y = W^ (Rn''E(A),E), p e (1, to).

Theorem 2.1 Suppose the Condition 2.2 holds. Assume E is a Banach space satisfying the weighted multiplier condition. Let A be a uniformly R-positive in E and X e SV2 with 0 < wa + + W2 < n. Then the problem (1.2) has a unique solution u and the coercive uniform estimate holds

J2 I X I^ \K * Da u\\X + \\A * u\\x + | X I \\u\\x < C\f \\x (2.7)

I a I <l

for allf e X and X e Sv.

Proof By applying the Fourier transform to equation (1.2) we get

«(£ )=D(f, X)f(f), D(f, X) = [A (f )+L(f ) + X]-1, ) = £ aa (£)(£ )a.

I a | <l

Hence, the solution of (1.2) can be represented as u(x) = F-1D(f, X)/ and there are positive constants C1 and C2 so that

CiIX | \\u\\x < ||F-1[ao(f,X)/]||X < C21 X |\\u\\x, Ci\A * u\\x < ||F-1[oi(£,X)f ]||x < C2\A * u\\x, C1J2 IX|1 ^ ||aa *Dau||x

< ||F-1[a2(f,X)/]||x < C2 £ IX|1 ^ ||aa *Dau||x, (2.8)

where a;(f, X) are operator functions defined in Lemma 2.1. Therefore, it is sufficient to show that the operator functions a;(f, X) are multipliers in x. However, this follows from Lemma 2.3. Thus, from (2.8) we obtain

IXI\\u\\x < Co\f \\x, \\A * u\\x < C1 \f \\x, £ IX11— ^ ||aa *Dau||x < C2\f \\x

for allf e X. Hence, we get the assertion.

Let O be an operator in X generated by problem (1.2) for k = 0, i.e. D(O) c Y, Ou =J2 aa * Da + A * u.

\a\<l D

Result 2.2 Theorem 2.1 implies that the operator O is separable in X, i.e. for all f e X there is a unique solution u e Y of the problem (1.2), all terms of equation (1.2) also are from X and there are positive constants C1 and C2 so that

CillOullx < J2 Ik *Dau\\X + IIA * u||X < C2||Ou||x.

Condition 2.3 Let D(A(x)) = D(A(£)), D(A(£)) is dense in E and does not depend on £; A(x) is uniformly positive in E. Moreover, there are positive constants C1 and C2 so that, for u e D(A), x e Rn,

Ci||A(£o)u|| < |\A(x)u|| < C2\|aa(£o)u||.

Remark 2.1 The Condition 2.3 is checked for the regular elliptic operators with smooth coefficients on sufficiently smooth domains ^ c Rm considered in the Banach space E = LP1(Q),p1 e (1, to) (see Theorem 4.1).

Theorem 2.2 Assume all conditions of Theorem 2.1 and Condition 2.3 are satisfied. Then forf e X and k e S((p) problem (1.2) has a unique solution u e Y and the coercive uniform estimate holds,

E1 IT II II

m1-T \\Dau\\X + \\Au\\x < C\\f \\x.

Proof By applying the Fourier transform to equation (1.2) we obtain D(£, k)ii(£) =/(£), where

D(£, k)=[A (£ )+L(£ ) + k]-1.

So, we see that the solution of equation (1.2) can be represented as u(x) = F-1D(f, k)f. Moreover, by the Condition 2.3 we have

||AF-1D(£,k)/||X < MlA(£o)F-1D(£,k)/||X.

Hence, by using the estimates (2.8) it is sufficient to show that the operator functions E\a\<l \k\1-^D(£,k) and A(£o)D(£,k) are multipliers in X. In fact, in view of (3) part

of Condition 2.2 and R-positivity of A these are proved by reasoning as in Lemma 2.3.

Condition 2.4 Let the Condition 2.3 hold and let there be positive constants Ci and C2 such that

Ci£ |â* m\l < \L(f )| < C2£ |a k m\l, f e Rn,

k=1 k=1

A(f )A-\x0) e Lœ(.Rn;B(E)), f,x0 e Rn,

QfA(x0)u|| < ||A(x)u|| < C2\A(x0)u||, u e D(A),x e Rn.

Theorem 2.3 Assume all conditions of Theorem 2.2 and Condition 2.4 are satisfied. Then for u e Y there are positive constants M1 and M2 so that

M1\\u\\Y < £ ||aa *Dau||x + \\A * u\\x < M2\u\Y.

Proof The left part of the above inequality is derived from Theorem 2.2. So, it remains to prove the right side of the estimate. In fact, from Condition 2.4 for u e Y we have

\\A * u\\x < M||F-1Au||x < C||F-1AA-1(xo)A(xo)u||x < C|F-1A(xo)u|x < C\\Au\\x.

Hence, applying the Fourier transform to equation (1.2) and by a reasoning as in Theorem 2.2, it is sufficient to prove that the function \a\<iOlafaEn=1 ft)-1 is a uniform multiplier in x. In fact, by using the Condition 2.4 and the proof of Lemma 2.3 we get the desired result. Consider the following example. □

Example 1 Let m = 2, n = 2, E = C, aa = a(x,y), A = b(x,y) such that a(f) and b(f) are positive real-valued functions for all f e R2 satisfying the Condition 2.2. Consider the equation

d 2u d 2u d2 u

-a * —- - a *--a * —-

d x2 d x d y d y2

+ b * u = f (x, y).

It is clear that the above equation satisfies all conditions of Theorem 2.1, i.e. the above p(

problem is Lp(R2) separable.

Result 2.3 Theorems 2.3 implies that for all u e Y there are positive constants C1 and C2 so that

C1\\u\\y < \\Ou\\x < C2\\u\\y.

From Theorem 2.1 we have the following.

Result 2.4 Assume all conditions of Theorem 2.1 hold. Then, for all X e Sv, the resolvent of operator O exists and the following sharp estimate holds:

£ x1-^ a * d(o+x)-1 |b(X)+\A * (o+x)-1 IB(X)

+ \\X(° + X)-1\\B(X) < C.

Result 2.5 Theorem 2.1 particularly implies that the operator O is uniformly sectorial in X; i.e. if A is uniformly R-positive for y e (2, n) then (see e.g. [29], Section 1.14.5) the operator O + a is a generator of an analytic semigroup in X.

From Theorems 2.1-2.3 and Proposition Ao we obtain the following.

Result 2.6 Let conditions of Theorems 2.1-2.3 hold for Banach spaces E e UMD, respectively. Then assertions of Theorems 2.1-2.3 are valid.

3 The Cauchy problem for parabolic CDOE

In this section, we shall consider the following Cauchy problem for the convolution parabolic equation:

— + ^ aa * Dau + A * u = f (t,x), u(0,x) = 0, (3.1)

91 rrt,

where A = A(x) is a possibly unbounded operator in a Banach space E, aa = aa(x) are complex-valued functions. For R++1, p = (p,pi), Z = Lp,Y (R++1;E) will denote the space of all p-summable E-valued functions with mixed norm (see e.g. [26], Section 4, for the complex-valued case), i.e., the space of all measurable E-valued functions f defined on R++1, for which

J (/ If (x, ¿C Y (x) dXjP dtY1 < TO.

Let E0 and E be two Banach spaces, where E0 continuously and densely embeds into E. Suppose I is an integer and WpY (R++1;E0, E) denotes the space of all functions u e Y such that we have the generalized derivatives Dtu,Dlku e Z, with the norm

Applying Theorem 2.1 we establish the maximal regularity of the problem (3.1) in the mixed norm Z. To this aim we need the following result.

Theorem 3.1 Suppose the Condition 2.2 holds, E is a Banach space that satisfies the uniform multiplier condition, and the operator A(£) is uniformly R-positive in E. Then the operator O is uniformly R-positive in X.

Proof Result 2.3 implies that the operator O is positive in X. We have to prove the R-boundedness of the set

a(£, À)-{X(O + X)"1: X e . From the proof of Theorem 2.1 we have X(O + X)-1/ - , Xf, f e X,

$(f, X) = X[A (f )+L(f ) + X]-1.

By definition of R-boundedness, it is enough to show that the operator function $(f, X) (dependent on the variable X and the parameter f) is a multiplier in X. In a similar manner to Lemma 2.3 one can easily show that $(f, X) is a multiplier in X. Then by the definition of R-boundedness we have

for all f1, f2,...,fm e Rn, X1, X2,...,Xm e Sy, f1,f2,...,fm e X, m e N, where {rj} is a sequence of independent symmetric {-1,1}-valued random variables on [0,1]. Hence, the set a (f, X) is uniformly R-bounded. □

From Theorem 3.1 and Proposition A0 we obtain the following.

Result 3.1 Let conditions of Theorem 3.1 hold for the Banach spaces E e UMD. Then the assertion of Theorem 3.1 is valid.

Now we are ready to state the main result of this section.

Theorem 3.2 Assume the Condition 2.2 holds for y e (n), E e UMD, and the operator A(f) is uniformly R-positive in E. Then for allf e Z equation (3.1) has a unique solution u e Wpu(R++1; E(A), E) satisfying

+ £\ua *Dau||Z + \\A * u\\z < C\\f \\z.

Z \a\<l

Proof It is clear that

Z = Lp1 (R+;X), WpY (R++1;E(A),E) = W1 (R+;D(O),X). Therefore, the problem (3.1) can be expressed as

^ + Ou(t) =f (t), u(0) = 0, t e R+. dt

By virtue of [1], Theorem 4.5.2, X e UMD provided E e UMD, p e (1, to). Then due to R-positivity of O with y e (n), by virtue of [1], Proposition 8.10, we see that for f e

Lpi(R+;X) equation (3.3) has a unique solution u e Wpi(R+;D(O),X) satisfying

+ II0uIIlw(r+x) < C\\f ||lw(r+x).

Lp1(R+X)

In view of Results 2.2, 2.6, from the above estimate we get (3.2). □

4 Degenerate convolution-elliptic equations

Consider the problem (1.1). Let

X = Lp(Rn;E), Y = WplY(Rn;E(A),E), p e (1, to). We show in this section the following result.

Theorem 4.1 Suppose the Condition 2.3 holds. Then for allf e X there is a unique solution of the problem (1.1) and the following coercive uniform estimate holds:

Y, m1-a Ik *D[a]u||X + ||A * ullX + m\\u\\X < C\f IIxx. (4.1)

Proof Let us make the following substitution:

yk - Y-1(z)dz, k - 1,2,...,n. Jo

It is clear that under the substitution (4.2), D[a]u transforms to Da u. Moreover, the spaces X and Y are mapped isomorphically onto the weighted spaces Lpy(Rn;E) and Wp~(Rn;E(A),E), respectively, where

Y = Y (y) = Y {x(y)) = y {xi(yi), X2(y2),...,Xn(yn)).

Moreover, under the substitution (4.2) the degenerate problem (1.1) is transformed into the following non-degenerate problem:

Y^aa * Da u + (A + X) * u = f, (.3)

considered in the weighted space Lpy(Rn; E) where

aa = aa(? (y)), u = u (y) = u( y (y)), A = (y)=A( ?(y)), f = f(y)=f{ 9(y)). Then in view of Theorem 2.1 we obtain the assertion. □

Now, we consider the Cauchy problem the degenerate parabolic convolution equation

— + ^ aa * D[a]u + A * u = f (t, x), (4.4)

d t —

u(0,x) = 0, t e R+,x e Rn. By using the map (4.2) we derive from Theorem 3.2 the following result.

Theorem 4.2 Assume the Condition 2.3 holds, for | < y < n, then problem (4.4) for all f e Lp(R++1; E) (p = (p, pi)) has a unique solution u(t, x) and the following estimate holds:

d u ~3t

Lp(R++1;E) |œ | —

+ £\aa * ^[aIu|ip(Rn+1;£) + 11-4 * U^Lp(R++1;£) - C\\f ^(R^E)-

Result 4.1 If we take |a| = 2, n = 2, then we see from Theorem 4.2 that the Cauchy problem

d [2]u

+ «11 * —+ «12 * d x2

d [2]u

9[2] u

„ „ + a22 * + - * u = f(t, x), u(0, x) = 0, (4-5) d X1 d x2 d x2

has a unique solution satisfying the coercive estimate

9 [2]u

d u ~3t

Lp(R+E) ij=1 for all f e Lp(R+; E).

« * 9x[1] 9xj1]

Lp(R+;E)

+ \\A * u|Lp(R+;E) - Clf \Lp(R+;E)

5 Boundary value problems for integro-differential equations

In this section by applying Theorem 2.1, the BVP for the anisotropic type convolution equations is studied. The maximal regularity properties of this problem in weighted mixed LPY norms are derived. In this direction we can mention e.g. [2,16], and [19].

Let i = Rn x i, where i c R^ is an open connected set with compact C2m-boundary 9i. Consider the BVP for integro-differential equation

(L + k)u = £ aa *Dau + £ (baVaD; + x) * u =f (x,y), x e Rn,y e Q,

la—l

Bju = £ bjß (y)Dß u(x, y) = 0, y e dQ, j = 1,2,..., m,

Ißl-mj

Dj = -i —, y = (y1,... ,yM), ba = ba (x), ^a = Va (y), 9 yj

aa = aa (x), a = (a1, a2,...,a„), aa = aa (x), u = u(x, y).

In general, l = 2m so, equation (5.1) is anisotropic. For l = 2m we get an isotropic equation. Let i = Rn x i, p = (p1, p), and y (x) = lxla, LP,Y (i) will denote the space of all p-summable scalar-valued functions with mixed norm (see e.g. [26], Section 4), i.e. the space of all measurable functions f defined on i, for which

\f IIlpy (i) = (/„(// (x, ^)|p1 Y (x) dx)p1 dyy < TO.

Analogously, WpY (i) denotes the weighted Sobolev space with corresponding mixed norm [26], Section 10. Let Q denote the operator generated by problem (5.1)-(5.2). In this section we present the following result.

Theorem 5.1 Let the following conditions be satisfied:

(1) na e C(fi) for each |a| = 2m and na e LTO(fi) + Lrk(fi) for each |a| = k < 2m with rk >pi,pi e (1,to), and 2m -k > l, va e LTO, -1 < a <p -1, k = 1,2,...,n;

(2) bjP e C2m-mj(9fi)for each j, ¡¡, ms < 2m, p e (1, to), X e Sv, y e [0, n);

(3)fory e fi, f e R^, a e Sw, n e (0, f), |f | + |a | =0 let a + ZM=2m na(y)fa =0;

(4) for each y0 e dfi localBVP in local coordinates corresponding to y0

a + J2 na(j0)Da#(y) = 0,

|a|=2m

Bj0# = J2 bjfi(y0)D(y) = hj, j = 1,2,...,m,

|3|=mj

has a unique solution $ e C0(R+) for all h = (h1, h2,...,hm) e Rm and for f' e R^-1 with |f '| + |X| =0;

(5) the (1) part of Condition 2.2 is satisfied, aa, ba e C(n)(Rn) and there are positive constants Ci, i = 1,2, so that

|f HD3 aa (f )| < C1, |f rD ba (f )| < C2 | b a (f )|,

f e Rn\{0}, 3k e{0,1},0 < |j01< n.

Then forf e LPY (fi) and X e Sy problem (5.1)-(5.2) has a unique solution u e Wp,y (¿2) and the following coercive uniform estimate holds:

E1 |a| II |i || ||

|X| -T ||aa * Dau|lp,y(jj)+ |||X|u|Lp,y(fi)

+ J2 \banaDa * u|lp,y(fi) < C\\f \Lp,Y(O).

|a|<2m

Proof Let E = Lp1(fi). It is well known [27] that Lp1(fi) is a UMD space for p1 e (1, to). Consider the operator A in Lp1 (fi) defined by

D(A) = Wpm(fi;Bju = 0), A(x)u = J^ ba(x)na(y)Dau(y). (5.3)

|a |<2m

Therefore, the problem (5.1)-(5.2) can be rewritten in the form of (1.2), where u(x) = u(x, •),f(x) =f (x, •) are functions with values in E = Lp1(fi). It is easy to see that A(f) and D3A (f) are operators in Lp1(fi) defined by

D(A) = D(D3A) = Wpm(fi;Bju = 0), A(f)u = £ ba(f)na(y)Dau(y),

|a|<2m

D^A(f)u = J2 d3ba(f)na(y)Dau(y).

|a|<2m

By virtue of [10], Theorem 4.1, we have

\ (A + ^)u|L„1(fi) < C|u|Wp21m(fi) < C\(A + ^)u|L„1(fi),

|| A + p)u\\Ln{Q) < C\\u\\W2,n(a) < CKD^A + p)u\\Ln{Q). Moreover, by using condition (5), for u e we have

if A + p)u\Ilpi(Q) < C||(A + p)u|^(Q).

Moreover, by [28] we see that the space Lpl(Q) satisfies the multiplier condition. Then all conditions of Theorem 2.1 hold and we obtain the assertion. □

6 Infinite system of IDEs

Consider the following infinite system of convolution equations:

y^ aa * Daum + £(dj + X) * uj(x)=fm(x), x e Rn, m = 1,2,________(6.1)

iai<l j=1

Condition 6.1 Let -1 < a <p -1, k = 1,2,..., n. There are positive constants C1 and C2 so that, for {dj(x)}f° e lq, for all x e Rn and some x0 e Rn,

C1|dj(x0)| < |dj(x)| < C2|dj(xo)|.

Suppose aa, dm e C(n)(Rn) and there are positive constants Mi, i = 1,2, so that

if iWiD aa (f)| < M1, if dm(f)| < M2|dm (f )|,

f e Rn\{0}, fa e {0,1}, 0 <iP i < n.

D(x) = { dm(x)}, dm >0, u = {um}, D * u = {dm * um},

lq(D) = ^u e lq, \\u\iq(D)= ^£dm(x0) * um\^ < to J, 1< q < to.

Here y (x)= ixia. Let Q be a differential operator in LPY (Rn; lq) generated by problem (6.1) and

B = B(LpY (Rn; lq)). Applying Theorem 2.1 we have the following.

Theorem 6.1 Suppose Condition 2.2(1) and Condition 6.1 are satisfied. Then:

(a) for allf (x) = fm(x)}TO e lpyY(Rn; lq(D)), for X e Sv, y e [0, n) problem (6.1) a unique solution u = {um(x)}TO that belongs to Wlp Y (Rn; lq(D), lq) and the coercive uniform estimate holds,

£ i X i 1-^ \\ aa * Dau|Lp,Y(Rn;lq) + \D * uKy (Rnlq)

ia i <l

+ i X i \\u\\lp,y(Rn;lq) < CWf \\lp,y(Rn;lq);

(b) for X e Sy there exists a resolvent (Q + X) 1 and

£ |X|1 ^ ||a„ * D(Q + X)-1 ]\b |a|<l

+ ||D * (Q + X)-1 ||B + ||X(Q + X)-1 ||B < C.

Proof In fact, let E = lq and A = [dm(x)Sjm], m, j = 1,2,..., to. Then

A (f ) = [dm (f )Sjm ], D3 A (f) = D dm (f j ], m, j = 1,2,..., to.

It is easy to see that A(f) is uniformly R-positive in lq and all conditions of Theorem 2.1 hold. Moreover, by [28] we see that the space lq satisfies the multiplier condition. Therefore, by virtue of Theorem 2.1 and Result 2.4 we obtain the assertions. □

Remark 6.1 There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1.1) and (3.1) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of A, byvirtue of Theorem 2.1 and Theorem 3.2 we can obtain the maximal regularity properties of different classes of convolution equations and Cauchy problems for parabolic CDEs or their systems, respectively.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details

1 Baku State University, Baku, Azerbaijan. 2Department of Mechanical Engineering, Okan University, Akfirat Beldesi, Tuzla, Istanbul, 34959, Turkey. 3 Department of Mathematics, Azerbaijan Khazar University, Baku, Azerbaijan.

Received: 14 December 2015 Accepted: 7 February 2016 Published online: 23 February 2016

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