For Shilov-type parabolic systems with nonnegative genus and coefficients of bounded smoothness, the properties of differentiability of Green’s function with respect to spatial variables are studied

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 1, pp. 365–379 DOI: 10.18514/MMN.2019.2089

SOME PROPERTIES OF GREEN’S FUNCTIONS OF SHILOV-TYPE PARABOLIC SYSTEMS

V. LITOVCHENKO AND G. UNGURYAN Received 15 September, 2016

Abstract. For Shilov-type parabolic systems with nonnegative genus and coefficients of bounded smoothness, the properties of differentiability of Green’s function with respect to spatial variables are studied.

2010Mathematics Subject Classification: 35K40; 35A08

Keywords: Cauchy problem, Shilov-type parabolic systems, Green’s function

1. INTRODUCTION

Developing the idea advanced by Ya.I. Zhitomirskii in [10] concerning the descrip- tion of the systems of partial differential equations parabolically steady to a change in coefficients, the following wide class of parabolic systems with variable coefficients was defined in [7]:

@tu.tIx/D fP0.tIi @x/CP1.t; xIi @x/gu.tIx/; .tIx/2˘_{.0IT }: (1.1) Here,uWDcol.u1; :::; um/; ˘.0IT WD f.tIx/W t2.0IT ; x2Rⁿg, andP0.tIi @x/and P1.t; xIi @x/are matrix differentiable expressions of the orders, respectively, pand p1,p > p10;with coefficients depending on the time variablet. In this case, the coefficients of the expressionP1can depend also on the spatial variablex. It is also assumed that the corresponding system

@tu.tIx/DP0.tIi @x/u.tIx/; .tIx/2˘.0IT ; (1.2) is uniformly parabolic by Shilov with parabolicity indexh,0 < hp, nonnegative genus;and reduced orderp₀[2]. In other words, there exist constantsı₀> 0and ı10such that, for allx2Rⁿandt2Œ0IT ;the following inequality holds:

.tIx/WD max

j2Nm

Rej.tIx/ ı0kxk^hCı1:

Here,j are roots of the equation det.P0.tI/ E/D0; 2Cⁿ; E is the identity matrix of the orderm,k kis the Euclid norm inRⁿ, andNmWD f1; 2; : : : ; mg.

We recall also that the reduced order p0 of a system is called the exact power order of growth of a function.I /in the complex spaceCⁿ(for a parabolic system,

c 2019 Miskolc University Press

we have always p0 > 1). In addition, the genus of a system parabolic by Shilov is called the largest index such that the function.I /in the domainG.K/WD Œ0IT  fxCiy2CⁿW kyk K.1C kxk/gwith someK > 0satisfies the estimate

.tIxCiy/ ıkxk^hCı1; ı> 0; ı10:

It is known that1 .p₀ h/1:

The following condition must be satisfied for (1.1):

(A)W0p1< h n.1 h=p0/ .m 1/.p h/:

It is obvious that the class of systems (1.1) envelops completely the class of Shilov- type parabolic systems with nonnegative genus. In this connection, it was proposed in [10] to call such systems Shilov-type parabolic systems with nonnegative genus and variable coefficients. To a certain extent, these systems solve the problem of proper extension of the notion of parabolicity by Shilov to the linear systems of equations with variable coefficients, since the Shilov class of systems is not parabolically stable to a change in its coefficients [4].

Let us consider systems (1.1) under the condition that their coefficients are the complex-valued functions that are continuous in the variable t, infinitely differen- tiable with respect to the variable x; and bounded together with their derivatives in a ball ˘Œ0IT : For these systems, Green’s functionZ.t; xI; /, 0 < t T, fx; g Rⁿ, was constructed, and its main properties such as the smoothness and the behavior in vicinities of infinitely remote spatial points were studied [7]. On the basis of these results, the theory of the Cauchy problem was developed in [5,8,9]

for such systems in spaces of the typeS by Gelfand and Shilov [3]. In particular, the proper solvability of the Cauchy problem with generalized initial data like the Gevrey ultradistributions was established, the form of a representation of classical solutions with generalized limiting values on the input hyperplane was obtained, their qualitat- ive properties were studied, and the sets of generalized input functions, for which the corresponding solutions are elements of the Schwartz spaceS or some space of the typeS;were described.

In order to get similar results under weaker conditions imposed on the coeffi- cients of system (1.1), it is necessary firstly to clarify the properties of corresponding Green’s function Z. In the present work, we will study properties of the function Z.t; xI; /with respect to spatial variablesxand under the minimum conditions of smoothness imposed on the coefficients of this system.

2. AUXILIARY INFORMATION

Let the differential expressionsP0andP1of system (1.1) have the structure P0.tIi @x/D X

jkjCp

A0;k.t /@^k_x; P1.t; xIi @x/D X

jkjCp₁

A1;k.tIx/@^k_x;

whereA_0;k.t /WDi^jkjC

a^lj_0;k.t /m

l;jD1,A_1;k.tIx/WDi^jkjC

a^lj_1;k.tIx/m

l;jD1are mat- rix coefficients,iis the imaginary unity, andjkjCWDk1C: : :Ckn; k2Zⁿ_C.

By G.t; I /, 0 < t T, we denote Green’s function of system (1.2). It is known that G.t; I /DF

^t./

.t; I /, where F Œis the Fourier transformation operator, and^t./is a matriciant of the corresponding Fourier duality of the system.

The following proposition is proper [5,6]:

8T > 09ı > 08k2Zⁿ_C9c > 08t2.IT 8 2Œ0IT /8 fx; g RⁿW j@^k_xG.t; Ix /j c.t /

nCjkjCC

h e ^ı

kx k .t /˛

₁¹_˛

; (2.1)

where WD.m 1/.p h/and˛WD=p0,j.a_lj/^m_l;jD1j WD max

fl;jgNm

ja_ljj.

Here, we consider systems (1.1), which satisfy, in addition to condition (A), the following condition:

(B): the coefficientsa^lj_0;k.t /,a^lj_1;k.tIx/are continuous in the variablet uni- formly with respect tox, differentiable with respect to the variablex up to the order˛inclusively, and bounded together with their deri- vatives by complex-valued functions in a ball˘Œ0IT .

In [7], Green’s function of system (1.1) was constructed in the form

Z.t; xI; /DG.t; Ix /CW .t; xI; /; .t; xI; /2˘_T²; (2.2) where˘_T² WD f.t; xI; /j0 < tT;fx; g Rⁿg,

W .t; xI; /WD

t

Z

dˇ Z

Rⁿ

G.t; ˇIx y/˚.ˇ; yI; /dy; (2.3) and

˚.t; xII/D

1

X

lD1

K_l.t; xI; /; (2.4)

where

K1.t; xI; /WDP1.t; xIi @x/G.; Ix /;

K_l.t; xI; /WD

t

Z

dˇ Z

Rⁿ

K1.t; xIˇ; y/K_{l 1}.ˇ; yI; /dy; l > 1: (2.5) In this case, it was established that condition (A) and the boundedness of the coef- ficients of system (1.1) ensure the absolute uniform convergence of the functional series (2.4) for allfx; g Rⁿ,t2.IT ;and 2Œ0; T /. Moreover, its sum˚ and the iterated kernelsKl satisfy the estimates

j˚.t; xI; /j c1.t /^{˛0 .1C˛ n/}e ^ı1

kx k .t /˛

₁¹_˛

; (2.6)

jK_l.t; xI; /j c₀^l 0

@

l 1

Y

jD1

c_.j"/B.˛0; j˛0/ 1 A

.t /^{l˛0 .1C˛ n/}e ı.1 .l 1/"/

kx k .t /˛

¹

1 ˛

; "2.0I1/; (2.7) with the estimating constants independent oft; ; x;and. Here˛0WD1C˛ n .nC p1C /= h > 0andB.;/is the Euler beta-function.

We note that estimates (2.1) and (2.6) forfx; g Rⁿand0 < tT guarantee the absolute convergence of the integral, by which the potentialW is determined.

Thus, the matrix functionZ.t; xI; /is properly determined by formula (2.2) on the whole set˘_T².

Completing this item, we present the following estimates from [1], which will be of importance in what follows:

e

ı (

kx yk .t ˇ/˛

¹

1 ˛

C ky k

.ˇ /˛

¹

1 ˛

)

e ^ı

kx k .t /˛

₁¹_˛

I (2.8)

Z

Rⁿ

e

ı (

kx yk .t ˇ/˛

₁¹_˛ C

ky k .ˇ /˛

₁¹_˛ )

dy

..t ˇ/.ˇ //^{˛ n}

c"e ^{ı.1 "/}

kx k .t /˛

¹

1 ˛

.t /^{˛ n} ; ı > 0 (2.9)

(here,fx; y; g Rⁿ,ˇ2.It /,0 < tT,"2.0I1/;andı > 0, and the quantity c"depends only on").

3. PROPERTIES OF GREEN’S FUNCTION

First, we estimate the derivatives of the iterated kernelsKl.

According to representation (2.5), the smoothness of the kernelK1.t; xI; /in the spatial variablesx and is determined, respectively, by the smoothness of the coefficients of system (1.1) and the functionG.t; Ix /. Therefore, there exist the derivatives@^r@^q_xK1forfr; qg Zⁿ_C,jqjC˛, and the following equality holds:

@^r@^q_xK1.t; xI; /D X

jkjCp1

q

X

lD0

C_q^l

@^l_xA_1;k.tIx/ @^k_{.x /}^{CrCq l}G.t; Ix /

; whereC_q^l is a binomial coefficient. From whence, with regard for condition (B) and estimate (2.1) forfr; qg Zⁿ_C,jqjC˛,.t; xI; /2˘_T²;we get

j@^r@^q_xK1.t; xI; /j cr;q.t /

nCp1CCjrCqjC

h e ^ı

kx k .t /˛

₁¹_˛

(3.1)

(here, the estimating constants are independent oft; ; x;and).

Forl > 1;we will use the representation Kl.t; xI; /D

t1

Z

dˇ Z

Rⁿ

K1.t; xIˇ; C/Kl 1.ˇ; CI; /d

C

t

Z

t1

dˇ Z

Rⁿ

K1.t; xIˇ; x ´/K_{l 1}.ˇ; x ´I; /d´; t1WDtC 2 : According to it,

@^r@^q_xK_l.t; xI; /D X

jr1jCjrjC

C_r^r1

t1

Z

dˇ Z

Rⁿ

@^r1@^q_xK₁.t; xIˇ; C/

@^{r r 1}K_{l 1}.ˇ; CI; /

dC X

jq1jCjqjC

C_q^q1

t

Z

t1

dˇ Z

Rⁿ

@^q_x¹K1.t; xIˇ; x ´/

@^r@^{q q}_x ¹K_{l 1}.ˇ; x ´I; /

d´; jqjC˛; .t; xI; /2˘_T²: (3.2) Hence, the estimation ofj@^r@^qxKl.t; xI; /jis reduced to that of the expressions

j@^r@^q_xK1.t; xI; C/j; j@^q_xK1.t; xI; x ´/j; j@^rK_{l 1}.t; CI; /j; j@^r@^q_xK_{l 1}.t; x ´I; /j:

In view of the boundedness of@^qxa^lj_1;k.tIx/,jqjC˛, and estimate (2.1), for all fq; rg 2Zⁿ_C,jqjC˛;fx; ; g 2Rⁿ; t2.IT ;and2Œ0IT /;we have

ˇˇ@^r@^q_xK1.t; xI; C/ˇ

ˇm X

jkjCp1

X

jq1jCjqjC

C_q^q1ˇ

ˇ@^q_x¹A_1;k.tIx/ˇ ˇ

ˇ

ˇ@^k_{.x /}^{CrCq q1}G.t; Ix /ˇ

ˇcr;q.t /

nCp1CCjrCqjC

h e ^{ı k}

x k .t /˛

₁¹_˛ I (3.3) ˇˇ@^q_xK1.t; xI; x /ˇ

ˇD ˇ ˇ

ˇ@^q_x X

jkj_Cp₁

A_1;k.tIx/@^k_xG.t; I/ˇ ˇ ˇm

ˇ ˇ

ˇ@^q_xA1;0.tIx/

ˇ ˇ ˇ

ˇ ˇ

ˇG.t; I/ ˇ ˇ

ˇbcq.t / ^nChe ^{ı k}

k .t /˛

₁¹_˛

cq.t /

nCp1C

h e ^{ı k}

k .t /˛

₁¹_˛ : (3.4)

We now estimate the expressionˇ

ˇ@^rKl.t; CI; /ˇ ˇ:Since

@^rK1.t; CI; /D X

jkjCp₁

@^rA1;k.tIC/@^kG.t; I/; .t; xI; /2˘_T²; (3.5) we have, according to condition (A), that the iterated kernels Kl.t; CI; /are differentiable with respect to the variable only to the order˛. This fact and (3.2) imply that@^qxKl.t; xI; /,jqj˛, is also a function differentiable with respect to only to this order˛.

Representation (3.5) and estimate (2.1) yield ˇˇ@^rK1.t; CI; /ˇ

ˇc1;r.t /

nCp1C

h e ^{ı k}

k .t /˛

₁¹_˛

: (3.6)

We note that

@^rK2.t; CI; /D@^r

t

Z

dˇ Z

Rⁿ

K1.t; CIˇ; y/K1.ˇ; yI; /dy

! : Let us change the variable of integration in the last integral by the formulayD´C: In view of estimates (3.6) and (2.9) and the equalities

t

Z

..t ˇ/.ˇ //^{˛0 1}dˇD.t /^{2˛0 1}B.˛0; ˛0/ (3.7) and

@^rK1.t; CI; ´C/D@^rK1.t; . ´/CI; / ˇ ˇ ˇ

D´C; we get

ˇˇ@^rK2.t; CI; /ˇ

ˇm X

jr1jCjrjC

C_r^r1

t

Z

dˇ Z

Rⁿ

ˇ ˇ

ˇ@^r1K1.t; CIˇ; ´C/ ˇ ˇ ˇ

ˇ ˇ

ˇ@^{r r 1}K1.ˇ; ´CI; / ˇ ˇ

ˇd´m X

jr1jCjrjC

C_r^r1c1;r1c_{1;.r r1/}

t

Z

.t ˇ/.ˇ / ^nCp1_h^C Z

Rⁿ

e ^ı

k ´k .t ˇ/˛

₁¹_˛

C _.ˇ^k´ _/˛^k ₁¹_˛ d´dˇ

c2;r."/B.˛0; ˛0/.t /^{˛0 nCp1hC}e ^{ı.1 "/ k}

k .t /˛

₁¹_˛

; "2.0I1/:

By reasoning analogously step by step, we arrive at the inequality ˇˇ@^rK_l.t; CI; /ˇ

ˇc_l;r."/^{l 1}Y

jD1

B.˛0; j˛0/

.t /^{.l 1/˛0}

nCp1C h

e ı.1 .l 1/"/ _.t^k _/˛^k ₁¹_˛

; (3.8)

which is satisfied for all f; g Rⁿ, jrjC ˛, 0 < t T, "2.0I1/; and l2Nn f1gand, hence, until the existence of such numberl, for which

ˇˇ@^rK_l.t; CI; /ˇ

ˇc_l;r."/^lY¹

jD1

B.˛0; j˛0/

e ^{ı.1 .l 1/"/ k}

k .t /˛

₁¹_˛ (3.9) (here, the quantitiesc_l;r."/ > 0do not depend on the variablest; ; ;and, which vary in the above-indicated way).

Since

@^r@^q_xK_l.t; xI; C/D@^r@^q_xK_l.t; xI; /ˇ ˇ ˇ

DC;

@^r@^q_xK_l.t; x ´I; /D@^r@^q_yK_l.t; yI; / ˇ ˇ ˇ_y

Dx ´;

the expressions @^r@^q_xK_l.t; xI; C/; @^r@^q_xK_l.t; x ´I; / and @^r@^q_xK_l.t; xI; / are of the same type. Therefore, with regard for representation (3.2) and estimates (3.3), (3.4), (3.8), and (2.9), we have

ˇˇ@^r@^q_xK2.t; xI; /ˇ

ˇm2^jrCqjC X

jr1jCjrjC

cr₁;qc_{1;.r r1/}

t1

Z

.t ˇ/

nCp1CCjr1CqjC h

.ˇ /

nCp1C h

Z

Rⁿ

e ^ı

kx k .t ˇ/˛

₁¹_˛

C _.ˇ^kk _/˛1¹_˛ ddˇ

C X

jq1jCjqjC

cq₁c_{r;.q q1/}

t

Z

t1

.ˇ /

nCp1CCjrCq q1jC

h .t ˇ/ ^nCp1hC Z

Rⁿ

e ^ı k´k

.t ˇ/˛

₁¹_˛

C ^kx_.ˇ ^´ _/˛^k₁¹_˛ d´dˇ

!

m2^jrCqjCc"e ^{ı.1 "/ k}

x k .t /˛

₁¹_˛

.t / ^{˛ n} X

jr1jCjrjC

cr₁;qc_{1;.r r1/}

t1

Z

.t ˇ/^{˛ n}

nCp1CCjr1CqjC

h .ˇ /^{˛0 1}dˇC X

jq₁jCjqjC

cq1cr;.q q₁/

t

Z

t₁

.t ˇ/^{˛0 1}.ˇ /^{˛ n}

nCp1CCjrCq q1jC

h dˇ

;jrjC˛;jqjC˛; "2.0I1/:

In view of the estimates

t1

Z

.t ˇ/^{˛ n}

nCp1CCjr1CqjC

h .ˇ /^{˛0 1}dˇ 2^jr1C

qjC

h .t /^{2˛0 1Cjr1C}

qjC h

B.˛0; ˛0/ and

t

Z

t1

.t ˇ/^{˛0 1}.ˇ /^{˛ n}

nCp1CCjrCq q1jC

h dˇ

2^j

rCq q1jC

h .t /^{2˛0 1C}

jrCq q1jC h

B.˛0; ˛0/;

we get the inequality ˇˇ@^r@^q_xK2.t; xI; /ˇ

ˇc_2;^r;q.t /^{2˛0 1C˛ nC}

jrCqjC h

B.˛0; ˛0/e ^{ı.1 "/ .tkx /˛k} ₁¹_˛

: By continuing stepwise the process of estimation, we obtain

ˇ

ˇ@^r@^q_xKl.t; xI; /ˇ

ˇc_l;"^r;q.t /^{l˛0 1C˛ nC}

jrCqjC h

e ı.1 .l 1/"/ _.t^kx _/˛^k₁¹_˛^{l 1}Y

jD1

B.˛0; j˛0/

; (3.10) for alljrjC˛;jqjC˛;fx; g Rⁿ; 0 < tT; "2.0I1/;andl2Nnf1g.

Let us pass to the estimation of the expressionˇ

ˇ@^r@^qxKl.t; xI; /ˇ

ˇ, which will be suitable for the establishment of the differentiability of the matrix function˚ with respect to the spatial variables. Directly from (3.10), we arrive at the existence of a numberlsuch that

ˇˇ@^r@^q_xK_l.t; xI; /ˇ

ˇc_l^r;q_;"e ^{ı.1 .l 1/"/ k}

x k .t /˛

₁¹_˛^l

1

Y

jD1

B.˛0; j˛0/ : Let us setl_CWDmaxfl; lg; l WDminfl; lg, wherelis the corresponding number from (3.9),"WD_r¹_lC; ıWDı 1 _r¹

; r> 2; T0WDmaxf1; Tg, and

c⁰WD max

l2N_l

Cnf1g

n

c1;r; c_l;r."/^{l 1}Y

jD1

B.˛0; j˛0/

; cr;q; c_l;"^{r;ql 1}Y

jD1

B.˛0; j˛0/o

;

cWDc⁰.T0/^{lC l} :Then (3.8) and (3.10) imply that, for allfx; ; g Rⁿ,0 <

tT,jrjC˛;andjqjC˛; ˇ

ˇ@^r@^q_xKl_C.t; xI; /ˇ

ˇce ^{ı k}

x k .t /˛

₁¹_˛

; ˇˇ@^rK_lC.t; CI; /ˇ

ˇce ^{ı .tkk /˛}

₁¹_˛ : In view of this result, estimate (2.8), the equality

Z

Rⁿ

e ^{ı0 .tkxˇ/˛yk}

₁¹_˛ dy .t ˇ/^{˛ n} D

Z

Rⁿ

e ^ı0k´k

1 1 ˛

d´DWbE <C1;

representation (3.2), and inequalities (3.3) and (3.4), we obtain ˇˇ@^rKl_CC1.t; CI; /ˇ

ˇ X

jr₁jCjrjC

C_r^r1

t

Z

dˇ Z

Rⁿ

ˇˇ@^r1K1.t; CIˇ; ´C/@^{r r 1}

K_lC.ˇ; ´CI; /ˇ

ˇd´mc² X

jr1jCjrjC

C_r^r1

t

Z

.t ˇ/^{˛0 1}

Z

Rⁿ

e ^ı

k ´k .t ˇ/˛

₁¹_˛

C _.ˇ^k´ _/˛^k ₁¹_˛ e

ı r

k ´k .t ˇ/˛

₁¹_˛ d´

.t ˇ/^n˛dˇ mc_r⁰bEc²B.˛0; 1/.t /^˛0e ^{ı k}

k .t /˛

₁¹_˛

; c_r⁰WD X

jr1jCjrjC

C_r^r1I

ˇ

ˇ@^r@^q_xKl_CC1.t; xI; /ˇ

ˇ X

jr1jCjrjC

C_r^r1

t1

Z

dˇ Z

Rⁿ

ˇ

ˇ@^r1@^q_xK1.t; xIˇ; C/

@^{r r 1}K_lC.ˇ; CI; /ˇ

ˇdC X

jq1jCjqjC

C_q^q1

t

Z

t1

dˇ Z

Rⁿ

ˇˇ@^q_x¹K1.t; xIˇ; x ´/

@^r@^{q q}_x ¹Kl_C.ˇ; x ´I; /ˇ ˇd´

mc² X

jr1jCjrjC

C_r^r1

t1

Z

.t ˇ/^{˛0 1Cjr1C}

qjC h

Z

Rⁿ

e ^ı

kx k .t ˇ/˛

₁¹_˛

C _.ˇ^k _/˛^k ₁¹_˛ e

ı r

kx k .t ˇ/˛

₁¹_˛ d .t ˇ/^{˛ n}dˇ

C X

jq1jCjqjC

C_q^q1

t

Z

t₁

t ˇ˛0 1Z

Rⁿ

e ^ı k´k

.t ˇ/˛

₁¹_˛

C ^k_.ˇ^{x ´} _/˛^k₁¹_˛

e ^{rı k}

´k .t ˇ/˛

₁¹_˛ d´

.t ˇ/^{˛ n}dˇ

!

mc²Eeb ^{ı k}

x k .t /˛

₁¹_˛

t

Z

.t ˇ/^{˛0 1}dˇ

X

jr1jCjrjC

C_r^r1.t t1/ ^jr1C

qjC h

! Ccq

!

mc²Eeb ^{ı .tkx /˛k} ₁¹_˛

.t /^˛0

B.˛0; 1/

2^j

rCqjC

h X

jr1jCjrjC

C_r^r1.t / ^jr1C

qjC h

! Ccq

!

mc_r;q⁰ c²E.2Tb 0/^j

rCqjC

h B.˛0; 1/.t /^{˛0 j}

rCqjC h e ^{ı k}

x k .t /˛

₁¹_˛

; c_r;q⁰ WDcrCcq: Applying the method of induction, we can verify firstly the validity of the estimate

ˇ

ˇ@^rKl_CCl.t; CI; /ˇ ˇ

c.mc⁰_rcbE.t /^˛0/^le ^{ı .tk /˛k}

₁¹_˛ ^{l 1} Y

jD1

B.˛0; 1Cj˛0/

!

; (3.11) and, hence, the estimate

ˇˇ@^r@^q_xK_lCCl.t; xI; /ˇ ˇc

mc_r;q⁰ cE.2Tb 0/^j

rCqjC h

l

.t /^{l˛0 j}

rCqjC h

e ^{ı k}

x k .t /˛

₁¹_˛ ^{l 1} Y

jD1

B.˛0; 1Cj˛0/

!

; (3.12) forjrjC˛,jqjC˛,.t; xI; /2˘_T² andl2Nnf1g.

The following propositions hold true.

Lemma 1. The matrix function˚.t; xI; /on the set˘_T² is a function differenti- able with respect to each of the spatial variablesxandto the order˛inclusively, and their derivatives satisfy the following estimates:

ˇˇ@^r@^q_x˚.t; xI; /ˇ

ˇc1.t /^{˛0 .1C˛ nCj}

rCqjC

h /e ^{ı .tkx /˛k} ₁¹_˛

; (3.13) ˇˇ@^r˚.t; CI; /ˇ

ˇc2.t /^{˛0 .1C˛ n/}e ^{ı .tk /˛k} ₁¹_˛

;f; g Rⁿ (3.14) (here, the estimating constantsc1; c2;andıare independent oft; ; x; ; ).

Proof. In any way, let us fix a point.x0I0/fromR²ⁿ;and let us consider a ball K^ı_.x

0I0/with radiusı > 0;which is centered at the point.x0I0/;in this space. Then, in view of structure (2.4) of the function˚ and the differentiability of the iterated kernels K_l with respect to spatial variables onR²ⁿ to the order ˛ inclusively, we can conclude that, in order to prove the differentiability of the matrix function˚ at the point .x0I0/ to the indicated order, it is necessary only to prove the uniform convergence of the formally differentiated series (2.4) in the variablesxand on the setK^ı_.x

0I0/,ı > 0(at every fixedt and; 0 < t T):

1

X

lD1

@^r@^q_xK_l.t; xI; /; jrjC˛;jqjC˛: (3.15) Directly from estimates (3.10) and (3.12) and the equality

l 1

Y

jD0

B.˛0; 1Cj˛0/D .˛₀/l

.1Cl˛0/;

where ./is the Euler gamma-function, forfr; qg Zⁿ_C,jrjC˛,jqjC˛, and .t; xI; /2˘_T²;we have

ˇ ˇ ˇ

1

X

lD1

@^r@^q_xKl.t; xI; / ˇ ˇ ˇ

l_C

X

lD1

ˇ ˇ

ˇ@^r@^q_xKl.t; xI; / ˇ ˇ ˇC

1

X

lDl_CC1

ˇ ˇ

ˇ@^r@^q_xKl.t; xI; / ˇ ˇ ˇ

c

l_C

X

lD1

.t /^{l˛0 .1C˛ nCj}

rCqjC h /

C

1

X

lD1

mc_r;q⁰ cbE.2T0/^j

rCqjC h l

.t /^{l˛0 j}

rCqjC h

^{l 1}Y

jD1

B.˛0; 1Cj˛0/

!

e ^{ı .tkx /˛k} ₁¹_˛

c1.t /^{˛0 1C˛ nC}

jrCqjC h

e ^{ı .tkx /˛k} ₁¹_˛

:

From whence, we get the uniform convergence of series (3.15) inxandand, hence, the validity of estimates (3.13).

Due to the corresponding estimates (3.8) and (3.11), we can verify analogously the validity of estimate (3.14).

The lemma is proved.

Lemma 2. The volumetric potentialW .t; xI; /on the set˘_T² is a function dif- ferentiable with respect to each of the spatial variablesxand to, respectively, the

orders˛Cp1and˛inclusively. In this case,

@^r@^q_xW .t; xI; /D

r

X

lD0

C_r^l

t1

Z

dˇ Z

Rⁿ

@^l@^q_xG.t; ˇIx y /@^{r l} ˚.ˇ; yCI; /dy

C

t

Z

t1

dˇ Z

Rⁿ

@^q_xG.t; ˇIx y/@^r˚.ˇ; yI; /dy; jqjCp1;jrjC˛; (3.16)

@^r@^q_xW .t; xI; /D

r

X

lD0

C_r^l

t1

Z

dˇ Z

Rⁿ

@^l@^q_xG.t; ˇIx y /@^{r l} ˚.ˇ; yCI; /dy

C

t

Z

t1

dˇ Z

Rⁿ

@^kG.t; ˇI/@^r@^{q k}_x ˚.ˇ; x I; /d; jrjC˛; (3.17) jkjCDp1; p1<jqjC˛Cp1:

Proof. ForjqjCp1andjrjC˛;we use the representation W .t; xI; /D

t1

Z

dˇ Z

Rⁿ

G.t; ˇIx y /˚.ˇ; yCI; /dy

C

t

Z

t1

dˇ Z

Rⁿ

G.t; ˇIx y/˚.ˇ; yI; /dy:

From whence, by the formal differentiation under the sign of integral, we obtain equality (3.16). Hence, in order to substantiate the validity of equality (3.16), it is sufficient to prove the uniform convergence of the following integrals in the variables xandonR²ⁿ:

I^r;l;q₁ .t1; xI; /WD

t1

Z

dˇ Z

Rⁿ

j@^l@^q_xG.t; ˇIx y /jj@^{r l} ˚.ˇ; yCI; /jdy;

jljC jrjCII₂^r;q.t; xIt1; /WD

t

Z

t1

dˇ Z

Rⁿ

j@^q_xG.t; ˇIx y/jj@^r˚.ˇ; yI; /jdy:

(3.18)

This convergence becomes obvious, if we take condition (A) and the following es- timates into account forfx; g Rⁿand0 < tT:

I^r;l;q₁ .t1; xI; /cc2Eeb ^{ı .tkx /˛k} ₁¹_˛

.t t1/

nCCjlCqjC h

t1

Z

.ˇ /^{˛0 1}dˇ;jljC jrjCI (3.19)

I₂^r;q.t; xIt1; / cc1Eeb ^{ı .tkx /˛k}

₁¹_˛

.t1 /

nCp1CCjrjC h

t

Z

t1

.t ˇ/^{˛0 1Cp1 j}

qjC

h dˇ: (3.20) These estimates follow directly from (2.1), (3.13), and (3.14).

We now prove the validity of formula (3.17). For this purpose, we fix anyk2Zⁿ_C such thatjkjCDp1. Then, according to (3.16) forp1<jqjC˛Cp1andjrjC

˛;we have

@^r@^q_xW .t; xI; /D

r

X

lD0

C_r^l@^{q k}_x

t1

Z

dˇ Z

Rⁿ

@^l@^k_xG.t; ˇIx y /@^{r l} ˚.ˇ; yCI; /

dyC@^{q k}_x

t

Z

t1

dˇ Z

Rⁿ

@^kG.t; ˇI/@^r˚.ˇ; x I; /d; .t; xI; /2˘_T²: Hence, it remains to substantiate the possibility to introduce the operation@^{q k}x under the signs of the corresponding integrals. In other words, we should prove the uniform convergence inxandof the following integrals onR²ⁿfor0 < tT:

t1

Z

dˇ Z

Rⁿ

@^l@^q_xG.t; ˇIx y /˚.ˇ; yCI; /dy;

t

Z

t1

dˇ Z

Rⁿ

@^kG.t; ˇI/@^r@^{q k}_x ˚.ˇ; x I; /d:

By reasoning similarly to the case of integrals (3.18) and using estimates (2.1), (3.13), and (3.14), we get the necessary convergence of the indicated integrals.

The lemma is proved.

The main result can be formulated as the following proposition.

Theorem 1. Let system(1.1)satisfy conditions (A) and (B). Then the correspond- ing function Z.t; xI; / defined by equality (2.2) is a function differentiable with respect to each of the spatial variablesx and on the set˘_T² to, respectively, the orders˛Cp1and˛inclusively, and

9ı > 08fr; qg Zⁿ_C;jqjC˛Cp1;jrjC˛;9c > 08.t; xI; /2˘_T² W j@^r@^q_xZ.t; xI; /j c.t /

nCjrCqjCC

h e ^{ı k}

x k .t /˛

₁¹_˛

I (3.21)

j@^kZ.t; xCI; /j c_k.t /^{ˇk nCh}e ^ı1 kxk

.t /˛

₁¹_˛

; (3.22)

where jkjC˛, 0 < t T, fx; g Rⁿ, ˇ_k WD

0; kD0;

˛0; k¤0 (here, the estimating constants are independent oft,,x;and).

Proof. With regard for structure (2.2) and the infinite differentiability of the func- tion G.t; I/ with respect to the variable , the smoothness of the function Z.t; xI; / in the variables x and becomes obvious directly from the assertion of Lemma2.

LetjqjCp1andjrjC˛. Then, according to (3.16), we get j@^r@^q_xZ.t; xI; /j j@^r_x ^CqG.t; Ix /j C

r

X

lD0

C_r^lI^r;l;q₁ .t1; xI; /CI^r;q₂ .t; xIt1; /:

From whence, by using estimates (2.1), (3.19), and (3.20), we obtain assertion (3.21).

In a similar way, by using formula (3.17), we verify the validity of assertion (3.21) also forp1<jqjC˛andjrjC˛.

Then, according to estimates (2.1) and (3.14), we have Y_k.t; xI; /WDˇ

ˇ ˇ

t

Z

dˇ Z

Rⁿ

G.t; ˇIx /@^k˚.ˇ; CI; /d ˇ ˇ ˇ

cc2 t

Z

.t ˇ/^{˛0Cp1h 1}.ˇ /^{˛0 1} Z

Rⁿ

exp n

ı0

nkx k .t ˇ/^˛

₁¹_˛

C kk .ˇ /^˛

₁¹_˛oo dydˇ

..t ˇ/.ˇ /^{˛ n}; ı0WDminfı; ıg;jkjC˛: Using estimate (2.9) and equality (3.7), we get

Yk.t; xI; /c".t /^{˛0 nCh}e ^{ı0.1 "/}

kxk .t /˛

₁¹_˛

; "2.0I1/;