DOI 10.1007/s10958-017-3536-8 Journal of Mathematical Sciences, Vol. 226, No. 3, October, 2017

EXISTENCE AND UNIQUENESS OF THE SOLUTION TO THE CAUCHY PROBLEM FOR THE STOCHASTIC REACTION-DIFFUSION DIFFERENTIAL EQUATION OF NEUTRAL TYPE A. N. Stanzhitskii1 and A. O. Tsukanova2

UDC 517.9

We prove a theorem on the existence and uniqueness of a mild solution to the Cauchy problem for a stochastic differential equation of neutral type in the weighted Hilbert space.

1. Introduction Numerous works are devoted to the problems of existence and uniqueness of the solutions of stochastic differential equations with given initial and boundary conditions in different function spaces [5, 7, 10–14] and, in particular, in Hilbert spaces [1, 3, 4, 8, 9]. Note that the nonlinear stochastic partial differential equations with delay (nonlinear stochastic differential equations of neutral type) and the properties of their solutions are of especial interest. The initial-value problem for an abstract functional equation of this type in a Hilbert space was considered and a theorem on existence and uniqueness of its mild solution was proved in [9]. However, it is difficult to check the conditions of this theorem in the general form for special applied problems. Thus, the problem of determination of the coefficient conditions for the existence and uniqueness of the solution, i.e., conditions expressed via the coefficients of the equation and, hence, convenient for verification, is of high importance. This can be done only in some special cases. The present paper is devoted to the analysis of one of these cases. The paper is organized as follows: The statement of the problem is presented in Sec. 2. Section 3 contains the required preliminary results. The main result of the paper is formulated in Sec. 4, and its proof is given in Sec. 5. In Sec. 6, we present a corollary. 2. Statement of the Problem Let .; F; P/ be a complete probability space. Consider the Cauchy problem for a stochastic reactiondiffusion integrodifferential equation of neutral type 0

B d @u.t; x/ C

Z

Rd

1

C b.t; x; u.˛.t /; ⇠/; ⇠/ d ⇠ A D .Åx u.t; x/ C f .t; u.˛.t //; x// dt C � .t; u.˛.t //; x/d W .t; x/;

1 Shevchenko 2 “Kyiv

u.t; x/ D �.t; x/;

�r  t  0;

x 2 Rd ;

0 < t  T;

x 2 Rd ;

(1)

r > 0;

Kyiv National University, Hlushkov Avenue, 4, Kyiv, 03680, Ukraine; e-mail: [email protected]. Polytechnic Institute” Ukrainian National Technical University, Pobeda Avenue, 37, Kyiv, 03056, Ukraine; e-mail: shugaray@

mail.ru. Translated from Neliniini Kolyvannya, Vol. 19, No. 3, pp. 408–430, July–August, 2016. Original article submitted February 29, 2016. 1072-3374/17/2263–0307

c 2017 �

Springer Science+Business Media New York

307

A. N. S TANZHITSKII AND A. O. T SUKANOVA

308

where T > 0 is a fixed number, Åx ⌘

d X

@2xi

iD1

is a d -dimensional Laplace operator, @2xi ⌘

@2 ; @xi2

i 2 f1; : : : ; d g; W .t; x/ is an L2 .Rd /-valued Q-Wiener process, ff; � gW Œ0; T ç ⇥ R ⇥ Rd ! R and bW Œ0; T ç ⇥ Rd ⇥ R ⇥ Rd ! R are given functions, �W Œ�r; 0ç ⇥ Rd ⇥  ! R are initial data, and ˛W Œ0; T ç ! Œ�r; 1/ is a function of delay. For problem (1), we prove the theorem on existence and uniqueness of a mild solution. 3. Preliminary Results In this section, we present the notation and some known results required in what follows. Assume that the flow of �-algebras fF t ; t � 0g is generated by an L2 .Rd /-valued Q-Wiener process W .t; x/ D

1 p X

�n en .x/ˇn .t /;

nD1

where fˇn .t/; n 2 f1; 2; : : :gg ⇢ R are independent Brownian motions, f�n ; n 2 f1; 2; : : :gg is a sequence of positive numbers such that 1 X

nD1

(2)

�n < 1;

and fen .x/; n 2 f1; 2; : : :gg is an orthonormal basis in L2 .Rd / such that sup n2f1;2;:::g

ess sup jen .x/j  1:

(3)

x2Rd

In what follows, we need some information from the theory of partial differential equations. Lemma 1 [2, p. 47]. If, in a homogeneous Cauchy problem @ t u.t; x/ D Åx u.t; x/; u.0; x/ D g.x/;

x 2 Rd ;

t > 0;

(4)

d

x2R ;

the initial data g belong to C.Rd / \ L1 .Rd /; then the solution of this problem can be represented in the form of a Poisson integral u.t; x/ D

Z

Rd

K.t; x � ⇠/g.⇠/ d ⇠;

(5)

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

309

where

K.t; x/ D

8 ˆ ˆ ˆ <

» x2 ; t > 0; exp � 4t º

1 d

.4⇡ t / 2

ˆ ˆ ˆ :

0;

t < 0;

is the heat-conduction kernel and, in addition, u 2 C 1 ..0; 1/ ⇥ Rd /: Lemma 2. If g belongs to L1 .Rd /; then function (5) satisfies the following limit relations: lim u.t; x/ D 0

jxj!1

and

lim @ t u.t; x/ D 0:

(6)

jxj!1

Proof. The proof of the lemma follows from the theorems on differentiability of the Lebesgue integral with respect to parameter and the possibility of limit transition in this integral. In what follows, the derivatives are understood in the ordinary sense. Lemma 3 [6, p. 319]. The derivatives of the kernel K admit the estimate j@rt @sx K.t; x/j

 cr;s t

� d2 �r� 2s

º

» c0 jxj2 exp � ; t

1 0 < c0 < : 4

cr;s > 0;

Lemmas 1–3 yield the following result: Lemma 4 [2, p. 360]. If g belongs to L1 .Rd / and jrx gj 2 L2 .Rd /;

kDx2 gk 2 L2 .Rd /;

(7)

then the second derivatives of function (5) admit the estimate sup

0t T

Z

2

.Åx u.t; x// dx D sup

0t T

Rd

Z

kDx2 u.t; x/k2 dx

C

Rd

Z

Rd

� � 2 �D g.x/�2 dx; x

(8)

� �T where C > 0 is a constant depending only on T; rx ⌘ @x1 : : : @xd ; 0

@2x1 B :: Dx2 ⌘ @ : @xd x1

1 : : : @x1 xd C :: :: A : : 2 : : : @xd

is the Hessian operator, and k � k is the corresponding norm of the matrix. Definition 1. A positive bounded function ⇢ 2 L1 .Rd / is called an admissible weight if, for every T > 0; there exists a constant C⇢ .T / > 0 such that the estimate Z

Rd

K.t; x � ⇠/⇢.x/ dx  C⇢ .T /⇢.⇠/;

⇠ 2 Rd ;

A. N. S TANZHITSKII AND A. O. T SUKANOVA

310

holds for any 0  t  T: Remark 1. The functions ⇢.x/ D expf�rjxjg;

r > 0;

⇢.x/ D

1 ; 1 C jxjr

r > d;

are typical examples of admissible weights. Remark 2. Without loss of generality, we assume that 0 < ⇢  1: ⇢

Here and in what follows, by L2 .Rd / we denote a weighted Hilbert space with admissible weight and the norm sZ kf kL⇢ .Rd / D 2

Rd

jf .x/j2 ⇢.x/ dx:

⇢

⇢

Lemma 5 [16; 17, p. 188]. The operators S.t /W L2 .Rd / ! L2 .Rd / generating the solution of problem (4) by the rule Z

u.t; x/ D .S.t /g.�//.x/ D

(9)

K.t; x � ⇠/g.�/ d ⇠

Rd

form a .C0 /-semigroup of operators with infinitesimal operator Åx : In this case, the inequality 2 2 k.S.t /g.�//.x/kL  C⇢ .T /kg.x/kL ; ⇢ ⇢ .Rd / .Rd / 2

2

0  t  T;

⇢

g 2 L2 .Rd /;

(10)

is true. ⇢

Let p � 2: By Bp;T;⇢ we denote the Banach space of all L2 .Rd /-valued random processes ˆ F t -measurable ⇢ for almost all 0  t  T W Œ0; T ç ⇥  ! L2 .Rd / and continuous in t for almost all ! 2  with the norm kˆkBp;T;⇢ D

r p

p

sup Ekˆ.t /kL⇢ .Rd / :

0t T

2

In the next section, we formulate a theorem on the existence and uniqueness of a mild solution of problem (1) for 0  t  T in the space Bp;T;⇢ : 4. Main Result In what follows, we assume that the following assumptions are true: (4.1) ˛W Œ0; T ç ! Œ�r; ˛.T /ç is a function from C 1 .Œ0; T ç/ with 0 < ˛ 0  1I (4.2) ff; � gW Œ0; T ç⇥R⇥Rd ! R and bW Œ0; T ç⇥Rd ⇥R⇥Rd ! R are functions measurable in the collections of their arguments;

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

311

(4.3) the initial function �W Œ�r; 0ç ⇥ Rd ⇥  ! R is F0 -measurable, independent of W .t; x/; t � 0; and such that p

sup Ek�.t /kL⇢ .Rd / < 1:

�rt0

(11)

2

Definition 2. A continuous random process uW Œ�r; T ç ⇥ Rd ⇥  ! R is called a mild solution of problem (1) if (i) it is F t -measurable for almost all �r  t  T I (ii) it is a solution of the integrodifferential equation

u.t; x/ D

Z

Rd

0

B K.t; x � ⇠/ @�.0/ C

1

Z

C b.0; ⇠; �.�r; ⇣/; ⇣/ d ⇣ Ad ⇠ �

Rd

�

Zt 0

C

0

B @Åx

Zt Z

Z

Rd

Z

b.t; x; u.˛.t /; ⇠/; ⇠/ d ⇠

Rd

0

B K.t � s; x � ⇠/ @

Z

Rd

1

1

C C b.s; ⇠; u.˛.s/; ⇣/; ⇣/ d ⇣ A d ⇠ Ads

K.t � s; x � ⇠/f .s; u.˛.s//; ⇠/ d ⇠ ds

0 Rd

C

Zt X 1 p 0 nD1

0

B �n @

Z

Rd

1

C K.t � s; x � ⇠/�.s; u.˛.s//; ⇠/en .⇠/ d ⇠ Adˇn .s/; 0  t  T;

u.t; x/ D �.t; x/;

x 2 Rd ;

�r  t  0;

x 2 Rd ;

(iii) it satisfies the condition E

ZT

p

ku.t /kL⇢ .Rd / dt < 1: 2

0

The following theorem is true for the solution thus defined: Theorem 1. Suppose that Assumptions 4.1–4.3 are true and, in addition,

r > 0I

A. N. S TANZHITSKII AND A. O. T SUKANOVA

312

(i) the functions ff; � g satisfy the condition of linear growth and the Lipschitz condition with respect to the second argument, i.e., there exists L > 0 such that jf .t; u; x/j C j�.t; u; x/j  L.1 C juj/;

0  t  T;

u 2 R;

x 2 Rd ;

jf .t; u; x/ � f .t; v; x/j C j� .t; u; x/ � � .t; v; x/j  Lju � vj; 0  t  T;

(12)

(13)

x 2 Rd I

fu; vg ⇢ R;

(ii) the function b satisfies the conditions

sup

0t T

0

Z

Rd

B @

sup

12

Z

Rd

0t T

C jb.t; x; 0; ⇠/jd ⇠ A ⇢.x/dx < 1;

Z Z

(14)

(15)

jb.t; x; 0; ⇠/jd ⇠dx < 1;

Rd Rd

and the Lipschitz conditions with respect to the third argument jb.t; x; u; ⇠/ � b.t; x; v; ⇠/j  l.t; x; ⇠/ju � vj;

0  t  T;

fx; ⇠g ⇢ Rd ;

fu; vg ⇢ R;

(16)

where the function lW Œ0; T ç ⇥ Rd ⇥ Rd ! Œ0; 1/ is such that sup

0t T

Z

Rd

0 B @

1

Z

l 2 .t; x; ⇠/ C d ⇠ A ⇢.x/dx < 1; ⇢.⇠/

Rd

Z v u Z l 2 .t; x; ⇠/ u sup d ⇠ dx < 1I t ⇢.⇠/ 0t T Rd

(17)

(18)

Rd

(iii) for any x 2 Rd ; the derivatives @xi b; @xi xj b; fi; j g ⇢ f1; : : : ; d g exist and, in addition, the gradient rx b and the matrix Dx2 b satisfy the condition of linear growth with respect to the third argument jrx b.t; x; u; ⇠/j C kDx2 b.t; x; u; ⇠/k  0  t  T;

d

.t; x; ⇠/.1 C juj/;

(19)

u 2 R;

fx; ⇠g ⇢ R ;

and the matrix Dx2 b satisfies the Lipschitz condition � 2 � �D b.t; x; u; ⇠/ � D 2 b.t; x; v; ⇠/�  x x 0  t  T;

d

fx; ⇠g ⇢ R ;

.t; x; ⇠/ju � vj;

fu; vg ⇢ R;

(20)

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

where the function

313

W Œ0; T ç ⇥ Rd ⇥ Rd ! Œ0; 1/ satisfies the conditions

sup

0t T

Z

0

Rd

sup

0t T

B @

Z

Rd

Z Z

12

C .t; x; ⇠/d ⇠ A dx < 1; 2 .t; x; ⇠/

⇢.⇠/

(21)

(22)

d ⇠ dx < 1I

Rd Rd

moreover, for any point x0 2 Rd ; there exists a neighborhood Bı .x0 / and a nonnegative function '.t; ⇠; x0 ; ı/ such that sup

0t T

'.t; �; x0 ; ı/ 2 L2 .Rd /; ı 2 RC ; p ⇢.�/

.t; x0 ; ⇠/j  '.t; ⇠; x0 ; ı/jx � x0 j;

j .t; x; ⇠/ �

0  t  T;

(23)

jx � x0 j < ı;

⇠ 2 Rd :

(24)

Under these conditions, if 0

B sup @

0t T

Z

Rd

0 B @

Z

Rd

1

1 p2

l 2 .t; x; ⇠/ C C d ⇠ A ⇢.x/dx A ⇢.⇠/

1

<

4p�1

(25)

;

then problem (1) possesses a unique mild solution u 2 Bp;T;⇢ on 0  t  T: 5. Proof of Theorem 1 The proof of the theorem is based on the classical Banach fixed-point theorem. Thus, we consider an operator ‰W Bp;T;⇢ ! Bp;T;⇢ :

.‰u/.t/ D

Z

Rd

0

B K.t; x � ⇠/ @�.0/ C

Z

Rd

�

Zt 0

C

0

B @Åx

Zt Z

0 Rd

Z

Rd

1

C b.0; ⇠; �.�r; ⇣/; ⇣/d ⇣ A d ⇠ �

Z

b.t; x; u.˛.t /; ⇠/; ⇠/d ⇠

Rd

0

B K.t � s; x � ⇠/ @

Z

Rd

1

1

C C b.s; ⇠; u.˛.s/; ⇣/; ⇣/d ⇣ A d ⇠ A ds

K.t � s; x � ⇠/f .s; u.˛.s//; ⇠/d ⇠ds

A. N. S TANZHITSKII AND A. O. T SUKANOVA

314

1 0 Zt X Z 1 p 4 X C B Ij .t /; �n @ K.t � s; x � ⇠/�.s; u.˛.s//; ⇠/en .⇠/d ⇠ A dˇn .s/ D C 0 nD1

j D0

Rd

0  t  T; u.t; x/ D �.t; x/;

x 2 Rd ; x 2 Rd ;

�r  t  T;

and prove that this operator is contracting. First, we show that ‰u belongs to Bp;T;⇢ for any u 2 Bp;T;⇢ : To this end, we estimate p

p

kIj .s/kBp;t;⇢ D sup EkIj .s/kL⇢ .Rd / ; 0st

j 2 f0; : : : ; 4g:

2

p

Further, we estimate kI0 .s/kBp;t;⇢ as follows:

p

kI0 .s/kBp;t;⇢

� 0 1 �p � �Z Z � � B C � � D sup E � K.s; x � ⇠/ @�.0/ C b.0; ⇠; �.�r; ⇣/; ⇣/d ⇣ A d ⇠ � � 0st � � ⇢ �Rd Rd

L2 .Rd /

0

� �p � Z � � � B � �  2p�1 B E K.s; x � ⇠/�.0/d ⇠ sup � � @0st � � � Rd � ⇢

L2 .Rd /

� 0 � Z Z � B � C sup E � K.s; x � ⇠/ @ .b.0; ⇠; �.�r; ⇣/; ⇣/ 0st � � Rd Rd p

�b.0; ⇠; 0; ⇣/ C b.0; ⇠; 0; ⇣//d ⇣/ d ⇠kL⇢ .Rd / 2

� �p �Z � � � � �  2p�1 sup E � K.s; x � ⇠/�.0/d ⇠ � � 0st � �Rd � ⇢

⌘

L2 .Rd /

� � Z � � C 4p�1 sup E � K.s; x � ⇠/ � 0st � Rd 0

B ⇥@

Z

Rd

�p � � C � .b.0; ⇠; �.�r; ⇣/; ⇣/ � b.0; ⇠; 0; ⇣//d ⇣ A d ⇠ � � � ⇢ 1

L2 .Rd /

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

315

� 0 1 �p � � Z Z � � B C � � K.s; x � ⇠/ @ b.0; ⇠; 0; ⇣/d ⇣ A d ⇠ � C 4p�1 sup E � � 0st � � ⇢ � Rd Rd

L2 .Rd /

D I01 C I02 C I03 :

By using (10) and (11), we can estimate the quantity I01 as follows: � �p � Z � � � � � K.s; x � ⇠/�.0/d ⇠ � I01 D 2p�1 sup E � � � 0st � � ⇢ Rd

p

p

 2p�1 C⇢2 .T /Ek�.0/kL⇢ .Rd / < 1: 2

L2 .Rd /

By virtue of (10), (11), (16), and (17) and the Cauchy–Bunyakovsky inequality, for the quantity I02 ; we find � 0 1 �p � � Z Z � � B C � � I02 D 4p�1 sup E � K.s; x � ⇠/ @ .b.0; ⇠; �.�r; ⇣/; ⇣/ � b.0; ⇠; 0; ⇣//d ⇣ A d ⇠ � � � 0st � � ⇢ Rd Rd

L2 .Rd /

� �p �Z � � � � �  4p�1 C⇢ .T /E � jb.0; x; �.�r; ⇣/; ⇣/ � b.0; x; 0; ⇣/jd ⇣ � � � �Rd � ⇢ p 2

L2 .Rd /

0

p B D 4p�1 C⇢2 .T /E @

0

Z

Rd

0

p B  4p�1 C⇢2 .T /E @

p B  4p�1 C⇢2 .T /E @

Z

p B D 4p�1 C⇢2 .T / @

Z

Rd

B @

Z

Rd

0

Rd

0

Rd

0

Z

Rd

0

B @

Z

B @

Z

Rd

0 B @

Z

Rd

12

1 p2

C C jb.0; x; �.�r; ⇣/; ⇣/ � b.0; x; 0; ⇣/jd ⇣ A ⇢.x/dx A 12

1 p2

p l.0; x; ⇣/ C C j�.�r; ⇣/j ⇢.⇣/d ⇣ A ⇢.x/dx A p ⇢.⇣/ l 2 .0; x; ⇣/ ⇢.⇣/

10

CB d ⇣A @

Z

Rd

1

1

1 p2

C C � 2 .�r; ⇣/⇢.⇣/ d ⇣ A ⇢.x/dx A 1 p2

0

l 2 .0; x; ⇣/ C C B d ⇣ A ⇢.x/dx A E @ ⇢.⇣/

Z

Rd

1 p2

C � 2 .�r; ⇣/⇢.⇣/d ⇣ A

A. N. S TANZHITSKII AND A. O. T SUKANOVA

316

0

p B D 4p�1 C⇢2 .T / @

0

Z

Rd

B @

1 p2

1

Z

Rd

l 2 .0; x; ⇣/ C C p d ⇣ A ⇢.x/dx A Ek�.�r/kL⇢ .Rd / < 1: ⇢.⇣/ 2

In view (10) and (14), for the quantity I03 ; we can write � 0 1 �p � �Z Z � � B C � � 3 p�1 sup E � K.s; x � ⇠/ @ b.0; ⇠; 0; ⇣/d ⇣ A dx � I0 D 4 � 0st � � ⇢ �Rd Rd

L2 .Rd /

� �p �Z � � � � �  4p�1 C⇢ .T / � jb.0; x; 0; ⇣/jd ⇣ � � � �Rd � p 2

0

p B D 4p�1 C⇢2 .T / @

Z

Rd

0 B @

Z

Rd

12

1 p2

C C jb.0; x; 0; ⇣/jd ⇣ A ⇢.x/dx A

< 1:

It follows from the established three estimates that p

(26)

kI0 .s/kBp;t;⇢ < 1: p

By using (11), (14), (16), (17), and the Cauchy–Bunyakovsky inequality, we estimate kI1 .s/kBp;t;⇢ as follows:

p

kI1 .s/kBp;t;⇢

� �p �Z � � � � � D sup E � b.s; x; u.˛.s/; ⇠/; ⇠/d ⇠ � � 0st � �Rd � ⇢

L2 .Rd /

0

B D sup E @ 0st

0

Z

Rd

0

Z

Rd

C

Z

Rd

0 B @

Z

Rd

Z

Rd

p B  2 2 sup E @

0st

B @

0 B @

12

1 p2

C C b.s; x; u.˛.s/; ⇠/; ⇠/d ⇠ A ⇢.x/dx A Z

Rd

12

C jb.s; x; u.˛.s/; ⇠/; ⇠/ � b.s; x; 0; ⇠/jd ⇠ A ⇢.x/dx 12

1 p2

C C jb.s; x; 0; ⇠/jd ⇠ A ⇢.x/dx A

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

0

B  2p�1 sup E @ 0st

0

Z

B @

0

B C 2p�1 sup @ 0st

B  2p�1 sup @ 0st

0

Z

Rd

B @

Rd

0

B C 2p�1 sup @ 0st

Z

Rd

0

B  2p�1 sup @ 0st

0

Z

Rd

B @

B C 2p�1 sup @ 0st

Z

Rd

0

Z

B @

12

1 p2

C C jb.s; x; 0; ⇠/jd ⇠ A ⇢.x/dx A 1 p2

1

0

l 2 .s; x; ⇠/ C C B d ⇠ A ⇢.x/dx A E @ ⇢.⇠/

Z

Rd

B @

Z

Rd

12

B @

Rd

C u2 .˛.s/; ⇠/⇢.⇠/d ⇠ A

C C jb.s; x; 0; ⇠/jd ⇠ A ⇢.x/dx A 1 p2

1

Z

1 p2

1 p2

l 2 .s; x; ⇠/ C C d ⇠ A ⇢.x/ dx A ⇢.⇠/ 0

Rd

Z

Rd

0

Z

1 p2

p l.s; x; ⇠/ C C ju.˛.s/; ⇠/j ⇢.⇠/d ⇠ A ⇢.x/dx A p ⇢.⇠/

0

Z

Rd

0

12

Z

Rd

Rd

317

p

sup Eku.˛.s//kL⇢ .Rd / 2

0st

12

1 p2

C C jb.s; x; 0; ⇠/jd ⇠ A ⇢.x/ dx A :

Let 0 < t ⇤ < ˛.T / be such that ˛.t ⇤ / D 0: Thus, we find p

p

p

sup Eku.˛.s//kL⇢ .Rd /  sup Eku.˛.s//kL⇢ .Rd / C sup Eku.˛.s//kL⇢ .Rd /

0st

0st ⇤

2

D 

2

p

sup Ek�.s/kL⇢ .Rd / C

�rs0

2

t ⇤ st

sup 0s˛.t /

p

2

p

Eku.s/kL⇢ .Rd / 2

p

sup Ek�.s/kL⇢ .Rd / C sup Eku.s/kL⇢ .Rd / < 1:

�rs0

2

0st

2

This yields p

kI1 .s/kBp;t;⇢ < 1:

(27)

In view of Remark 2, the Cauchy–Bunyakovsky inequality, and the Fubini theorem, we can estimate p kI2 .s/kBp;t;⇢ as follows:

A. N. S TANZHITSKII AND A. O. T SUKANOVA

318

p

kI2 .s/kBp;t;⇢

� 0 �Zs Z � � B K.s � ⌧; x � ⇠/⇥ D sup E � @Åx 0st � �0 Rd 0

Z

B ⇥@

Rd

0

B D sup E @ 0st

0

B ⇥@

Z

Rd

�p � � C C � b.⌧; ⇠; u.˛.⌧ /; ⇣/d ⇣ A d ⇠ A d ⌧ � � � ⇢ 1

1

L2 .Rd /

0

Z

B @

Rd

Zs 0

0

B @Åx

Z

K.s � ⌧; x � ⇠/

Rd

1

1

12

1 p2

C C C C b.⌧; ⇠; u.˛.⌧ /; ⇣/; ⇣/d ⇣ A d ⇠ A d ⌧ A ⇢.x/dx A

0 0 Zs Z Z B B sup E @ K.s � ⌧; x � ⇠/ @Åx

p

t2

0st

0

B ⇥@

Z

Rd

0 Rd

Rd

1

12

1 p2

C C C b.⌧; ⇠; u.˛.⌧ /; ⇣/; ⇣/d ⇣ A d ⇠ A dxd ⌧ A

�2 � 0 1 p2 � Zt Z � Z � � p p B C � � 2  C 2t 2E@ b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣ � dxd ⌧ A �Dx � � � 0 Rd � Rd

provided that the conditions of Lemma 4 with

g.⌧; x/ D

Z

b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣

Rd

and u.⌧; x/ D

Z

Rd

0

B K.s � ⌧; x � ⇠/ @

Z

Rd

1

C b.⌧; ⇠; u.˛.⌧ /; ⇣/; ⇣/d ⇣ A d ⇠

are satisfied. We now check the validity of Lemma 4 for this function. To this end, we prove that Z b.⌧; �; u.˛.⌧/; ⇣/; ⇣/d ⇣ 2 L1 .Rd / for any 0  ⌧  t with probability 1; (i) Rd

(28)

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

319

(ii) condition (7) is satisfied. 1. Indeed, by virtue of (11), (15), (16), (18), and the Cauchy–Bunyakovsky inequality, we obtain ˇ ˇ ˇ Z ˇˇ Z ˇ ˇ ˇ b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣ ˇ dx E ˇ ˇ ˇ ˇ ˇ Rd Rd E

Z Z

Rd Rd

Z Z p l.⌧; x; ⇣/ ju.˛.⌧ /; ⇣/j ⇢.⇣/d ⇣ dx C jb.⌧; x; 0; ⇣/jd ⇣ dx p ⇢.⇣/ Rd Rd

1 Z v u Z l 2 .⌧; x; ⇣/ r u C B 2 d ⇣ dx A sup Eku.˛.⌧//kL  @ sup t ⇢ d ⇢.⇣/ 2 .R / 0⌧ t 0⌧ t 0

Rd

C sup

0⌧ t

Rd

Z Z

jb.⌧; x; 0; ⇣/j d ⇣dx

Rd Rd

1 Z v u Z l 2 .⌧; x; ⇣/ r u C B 2 2 C sup Eku.⌧/kL sup Ek�.⌧ /kL d ⇣dx A  @ sup t ⇢ ⇢ d .Rd / ⇢.⇣/ 2 2 .R / 0⌧ t �r⌧ 0 0⌧ t 0

Rd

C sup

0⌧ t

Rd

Z Z

jb.⌧; x; 0; ⇣/jd ⇣dx < 1:

Rd Rd

With probability 1, this yields ˇ ˇ ˇ Z ˇˇ Z ˇ ˇ ˇ ˇ b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣ ˇ dx < 1: ˇ ˇ ˇ ˇ Rd Rd

2. We now prove the differentiability of the function g for x D x0 ; which is an arbitrary point from Rd : Let Bı .x0 / be the neighborhood from assertion (iii). Thus, by virtue of (19) and (24), we get jrx b.⌧; x; u.˛.⌧/; ⇣/; ⇣/j 

.⌧; x; ⇣/.1 C ju.˛.⌧/; ⇣/j/

 .j .⌧; x; ⇣/ �

.⌧; x0 ; ⇣/j C

 .'.⌧; ⇣; x0 ; ı/jx � x0 j C  .ı'.⌧; ⇣; x0 ; ı/ C

.⌧; x0 ; ⇣//.1 C ju.˛.⌧/; ⇣/j/

.⌧; x0 ; ⇣//.1 C ju.˛.⌧/; ⇣/j/

.⌧; x0 ; ⇣//.1 C ju.˛.⌧/; ⇣/j/:

We now show that .ı'.⌧; �; x0 ; ı/ C

.⌧; x0 ; �//.1 C ju.˛.⌧/; �/j/ 2 L1 .Rd /:

A. N. S TANZHITSKII AND A. O. T SUKANOVA

320

By virtue of the Cauchy–Bunyakovsky inequality and relations (11) and (21)–(23), we obtain E

Z

.ı'.⌧; ⇣; x0 ; ı/ C

.⌧; x0 ; ⇣//.1 C ju.˛.⌧/; ⇣/j/d ⇣

Rd

Dı

Z

Rd

'.⌧; ⇣; x0 ; ı/ p ⇢.⇣/d ⇣ C p ⇢.⇣/

Z

.⌧; x0 ; ⇣/d ⇣

Rd

C ıE

Z p '.⌧; ⇣; x0 ; ı/ ju.˛.⌧ /; ⇣/j ⇢.⇣/d ⇣ C E p ⇢.⇣/

Z

Rd

Rd

vZ vZ Z u ' 2 .⌧; ⇣; x ; ı/ u u 0 d⇣ u ⇢.⇣/d ⇣ C  ıt t ⇢.⇣/ Rd

Rd

vZ vZ u u u B u ' 2 .⌧; ⇣; x0 ; ı/ C @ı t d⇣ C t ⇢.⇣/ Rd

2 .⌧; x ; ⇣/ 0

⇢.⇣/

Rd

vZ Z u ' 2 .⌧; ⇣; x ; ı/ v uZ u 0  ıt ⇢.⇣/ d ⇣ C d ⇣u t ⇢.⇣/ Rd

vZ vZ u u u B u ' 2 .⌧; ⇣; x0 ; ı/ C @ı t d⇣ C t ⇢.⇣/ Rd

r

1

C d ⇣A

r

2 sup Eku.˛.⌧//kL ⇢ .Rd /

0⌧ t

2

.⌧; x0 ; ⇣/d ⇣

Rd

0

⇥

.⌧; x0 ; ⇣/d ⇣

Rd

0

Rd

p .⌧; x0 ; ⇣/ ju.˛.⌧ /; ⇣/j ⇢.⇣/d ⇣ p ⇢.⇣/

2 .⌧; x ; ⇣/ 0

⇢.⇣/

Rd

1

C d ⇣A

2 2 sup Ek�.⌧ /kL C sup Eku.⌧/kL < 1: ⇢ ⇢ .Rd / .Rd /

�r⌧ 0

2

0⌧ t

2

With probability 1, this leads to the required condition Z

.ı'.⌧; ⇣; x0 ; ı/ C

.⌧; x0 ; ⇣//.1 C ju.˛.⌧/; ⇣/j/d ⇣ < 1:

Rd

Hence, by the local theorem on differentiability of an integral with respect to the parameter, there exists rx g.⌧; x/ and the following equality is true: rx

Z

Rd

We now show that

b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣ D

Z

Rd

rx b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣:

(29)

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

rx

Z

321

b.⌧; �; u.˛.⌧/; ⇣/; ⇣/d ⇣ 2 L2 .Rd /:

Rd

By virtue of (11), (19), (21), (22), (29), and the Cauchy–Bunyakovsky inequality, we get ˇ2 ˇ ˇ Z ˇˇ Z ˇ ˇ ˇ b.⌧; x; u.˛.⌧/; ⇣/; ⇣/ d ⇣ ˇ dx E ˇrx ˇ ˇ ˇ ˇ Rd Rd E

Z

Rd

E

Z

Rd

2

Z

Rd

0 B @

Z

Rd

0 B @

C jrx b.⌧; x; u.˛.⌧/; ⇣/; ⇣/jd ⇣ A dx 12

Z

C .⌧; x; ⇣/.1 C ju.˛.⌧/; ⇣/j/d ⇣ A dx

Rd

0 B @

12

Z

Rd

 2 sup

0⌧ t

Z

Rd

0⌧ t

Rd

0

0 B @

0

0⌧ t

Rd

C .⌧; x; ⇣/d ⇣ A dx

Z Z

2 .⌧; x; ⇣/

⇢.⇣/

Rd Rd

0 B @

Z

Rd

0⌧ t

 2 sup

2 .⌧; x; ⇣/

⇢.⇣/

0 B @

C 2 d ⇣dx A Eku.˛.⌧//kL ⇢ .Rd / 2

C 2 d ⇣dx A Eku.˛.⌧//kL ⇢ .Rd /

C .⌧; x; ⇣/d ⇣ A dx 2 .⌧; x; ⇣/

⇢.⇣/

Rd Rd

Rd

1

2

12

Z Z

Z

1

12

Z

Rd

B C 2 @ sup Z

Z Z

Rd Rd

0⌧ t

Z

0

C B .⌧; x; ⇣/d ⇣ A dx C 2 @

B C 2 sup @  2 sup

12

12

1

C 2 d ⇣dx A sup Eku.˛.⌧//kL ⇢ .Rd / 0⌧ t

0

C B .⌧; x; ⇣/d ⇣ A dx C 2 @ sup

0⌧ t

2

Z Z

Rd Rd

2 .⌧; x; ⇣/

⇢.⇣/

1

C d ⇣ dx A

A. N. S TANZHITSKII AND A. O. T SUKANOVA

322

2 2 sup Ek�.⌧ /kL C sup Eku.⌧/kL ⇢ ⇢ .Rd / .Rd /

⇥

�r⌧ 0

0⌧ t

2

2

!

< 1;

whence, with probability 1, we arrive at the required condition

For (28),

Dx2

Z

Rd

p

ˇ ˇ2 ˇ Z ˇˇ Z ˇ ˇ ˇ b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣ ˇ dx < 1: ˇrx ˇ ˇ ˇ Rd ˇ Rd

b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣; condition (7) is proved in a similar way. This implies that, in inequality

kI2 .s/kBp;t;⇢

�2 � 0 1 p2 � Zt Z � Z � � p p B � C � 2  C 2t 2E@ b.⌧; x; u.˛.⌧/; ⇣/; ⇣/d ⇣ � dxd ⌧ A �Dx � � � 0 Rd � Rd

0 1 p2 12 0 Zt Z Z � C p p B C B � 2  C 2t 2E@ @ �Dx b.⌧; x; u.˛.⌧/; ⇣/; ⇣/� d ⇣ A dxd ⌧ A 0 Rd

Rd

0 0 Zt Z Z p p p B B 2 2 2  2 C t E@ @ 0 Rd

0

B C@

Zt Z Z

C .⌧; x; ⇣/d ⇣ A dxd ⌧

Rd

2 .⌧; x; ⇣/

⇢.⇣/

0 Rd Rd

p

p

 2p�1 C 2 t 2

0

0 0 Z B Zt Z BB B B@ @ @ 0 Rd

0 Zt Z Z B C@

2 .⌧; x; ⇣/

⇢.⇣/

sup

0⌧ t

Z

Rd

Z

Rd

1

1 p2

C C 2 d ⇣dxd ⌧ A ku.˛.⌧//kL ⇢ .Rd / A 2

12

Rd

0 Rd Rd

⇥

12

1 p2

C C .⌧; x; ⇣/d ⇣ A dxd ⌧ A 1 p2

1

C C p p�1 p d ⇣dxd ⌧ A Eku.˛.⌧//kL⇢ .Rd / C C 2 tp A2

.⌧; x; ⇣/d ⇣

!2

dx

2

! p2

C sup

0⌧ t

Z Z

Rd Rd

2 .⌧; x; ⇣/

⇢.⇣/

d ⇣dx

! p2

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

p

323

p

sup Ek�.⌧ /kL⇢ .Rd / C sup Eku.⌧/kL⇢ .Rd /

⇥

�r⌧ 0

0⌧ t

2

2

!!

(30)

< 1:

p

By using (10)–(12), we estimate kI3 .s/kBp;t;⇢ as follows: � �p � Zs Z � � � � � p K.s � ⌧; x � ⇠/f .⌧; u.˛.⌧ //; ⇠/d ⇠d ⌧ � kI3 .s/kBp;t;⇢ D sup E � � 0st � � 0 Rd � ⇢

L2 .Rd /

0

B D sup E @ 0st

12 1 p2 0 Zs Z C C B K.s � ⌧; x � ⇠/f .⌧; u.˛.⌧ /; ⇠/; ⇠/d ⇠d ⌧ A ⇢.x/dx A @

Z

0 Rd

Rd

0

p B  t 2 sup E @

0st

p

 Lp t 2

Zs Z

0 Rd

0st

Z

0 Rd

p

p

D 2 2 Lp t 2

C

Zs 0

1 p2

C C K.s � ⌧; x � ⇠/f .⌧; u.˛.⌧/; ⇠/; ⇠/d ⇠ A ⇢.x/dxd ⌧ A

Rd

0 Rd

0

p p B  2 2 Lp t 2 sup E @

C

B @

12

Z

0 12 1 p2 0 Zs Z Z B C C B sup E @ @ K.s � ⌧; x � ⇠/.1 C ju.˛.⌧/; ⇠/j/d ⇠ A ⇢.x/dxd ⌧ A

0st

Zs

0

0 B @

Z

Rd

Rd

Zs Z

0 Rd

0 B @

Z

Rd

12

C K.s � ⌧; x � ⇠/d ⇠ A ⇢.x/dxd ⌧ 12

1 p2

C C K.s � ⌧; x � ⇠/ju.˛.⌧/; ⇠/jd ⇠ A ⇢.x/dxd ⌧ A

0 Zs Z B sup E @ ⇢.x/dxd ⌧

0st

0 Rd

�2 � � �Z � � � � � K.s � ⌧; x � ⇠/ju.˛.⌧//jd ⇠ � � � � ⇢ �Rd

L2 .Rd /

1 p2

C d⌧C A

A. N. S TANZHITSKII AND A. O. T SUKANOVA

324

p

 2p�1 Lp t 2

0

1 p2 0 Zs Z B C B sup B ⇢.x/dxd ⌧ A @0st @ 0 Rd

0

�2 � � Zs � Z � B � � � K.s � ⌧; x � ⇠/ju.˛.⌧//jd ⇠ C sup E B � � @ � � 0st � ⇢ � 0 Rd

L2 .Rd /

0

0

Rd

p�1

2

p

L t

p 2

0

0

Rd

0

0

p B p B 2 D 2p�1 Lp t 2 B @t @

2

0

1 p2

Z

B pB Bt 2 @ @

1 0 t 1 p2 Z p C C 2 ⇢.x/dx A C C⇢2 .T /E @ ku.˛.⌧//kL d⌧A C ⇢ A .Rd / 1 p2

Z

p B p B 2  2p�1 Lp t 2 B @t @

1 p2 1 C C C d⌧C A C A

Z

Rd

p p�2 C ⇢.x/dx A C C⇢2 .T /t 2 E

Zt 0

1

C p ku.˛.⌧//kL⇢ .Rd / d ⌧ C A 2

1 p2

p p�2 C ⇢.x/dx A C C⇢2 .T /t 2

0 ⇤ 11 Zt Zt B CC p p ⇥ E @ ku.˛.⌧//kL⇢ .Rd / d ⌧ C ku.˛.⌧//kL⇢ .Rd / d ⌧ AA 2

0

0

p B p B 2 D 2p�1 Lp t 2 B @t @

Z

Rd

0

B ⇥E @

Z0

�r

2

t⇤

0

1 p2

p p�2 C ⇢.x/dx A C C⇢2 .T /t 2

11 ˛.t / Z 1 1 CC p p k�.˛.⌧//kL⇢ .Rd / 0 ku.˛.⌧//kL⇢ .Rd / 0 d˛.⌧ /AA : d˛.⌧ / C ˛ .⌧ / ˛ .⌧ / 2 2 0

It follows from Proposition 4.1 that there exists c > 0 such that 1 ˛ 0 .⌧ / Hence,

 c;

0  ⌧  t:

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

2

p�1

p

L t

p 2

0

0

B pB Bt 2 @ @

1 p2

Z

Rd

0

p 2

p�2 C ⇢.x/dx A C C⇢ .T /t 2 E @

Z0

�r

325

1

p

k�.˛.⌧//kL⇢ .Rd / 2

˛ 0 .⌧ /

d˛.⌧ /

11 ˛.t/ Z 1 CC p C ku.˛.⌧//kL⇢ .Rd / 0 d˛.⌧ /AA ˛ .⌧ / 2 0

0

0

p B p B 2  2p�1 Lp t 2 B @t @

Z

Rd

0

B ⇥E @

Z0

�r

p p�2 C ⇢.x/dx A C cC⇢2 .T /t 2

11 ˛.t / Z CC p p k�.⌧ /kL⇢ .Rd / d ⌧ C ku.⌧ /kL⇢ .Rd / d ⌧ AA 2

2

0

0

0

p B p B 2  2p�1 Lp t 2 B @t @

Z

Rd

p 2

CcC⇢ .T /t

p�2 2

00

BB  2p�1 Lp t p B @@ ⇥

1 p2

sup

�r⌧ 0

Z

Rd

r

1 p2

C ⇢.x/dx A sup

�r⌧ 0

p Ek�.⌧ /kL⇢ .Rd / 2

C ˛.t /

sup

0⌧ ˛.t /

p Eku.⌧/kL⇢ .Rd / 2

!!

1 p2

p C ⇢.x/dx A C cC⇢2 .T /

p Ek�.⌧ /kL⇢ .Rd / 2

C sup

0⌧ t

p Eku.⌧/kL⇢ .Rd / 2

!!

< 1:

(31)

p

Further, we estimate kI4 .s/kBp;t;⇢ for p > 2: By virtue of Lemma 7.2 in [18, p. 182], we obtain � s �p �Z � � � � E � S.s � ⌧ /�.⌧; u.˛.⌧ //; ⇠/d W .⌧; x/� � � � 0

0 s 1 p2 Z 2 A :  Cp E @ kS.s � ⌧ /�.⌧; u.˛.⌧//; ⇠/kL 0d ⌧ 2

0

(32)

A. N. S TANZHITSKII AND A. O. T SUKANOVA

326

Here, k � kL0 is the corresponding Hilbert–Schmidt norm appearing in the structure of the stochastic integral over 2 the Q-Wiener process [18, p. 91]. In this case, inequality (32) takes the form � s �p �Z � � � � E� � S.s � ⌧/�.⌧; u.˛.⌧ //; ⇠/d W .⌧; x/� � � 0

0 12 1 1 p2 0 0 Zs X Z Z 1 B C C C B B  Cp E @ �n @ @ K.s � ⌧; x � ⇠/�.⌧; u.˛.⌧ /; ⇠/; ⇠/en .⇠/d ⇠ A ⇢.x/dx A d ⌧ A 0 nD1

Rd

Rd

0 0 1 1 p2 s 1 Z Z p B B X C C  C⇢2 .T /Lp Cp E @ �n @ .1 C ju.˛.⌧/; x/j/2 en2 .x/⇢.x/dx A d ⌧ A 0 nD1

1 X

p

 22

�n

nD1

! p2

Rd

0 1 p2 s Z s Z Z p B C 2 C⇢2 .T /Lp Cp @ ⇢.x/dxd ⌧ C ku.˛.⌧//kL d⌧A : ⇢ .Rd / 2

0 Rd

0

The expression

2

p 2

1 X

�n

nD1

! p2

p

C⇢2 .T /Lp Cp

is denoted by A: Hence, the last expression does not exceed the following expression:

2

p�2 2

0

p B At 2 @

Z

Rd

2

1 p2

0

p�2 C ⇢.x/dx A C 2 2 A @

p�2 2

At

p 2

0 B @

Z

Rd

Zs 0

1 p2

2 d⌧A ku.˛.⌧//kL ⇢ .Rd / 2

1 p2

p�2 p�2 C ⇢.x/dx A C 2 2 At 2

Zs

p

ku.˛.⌧//kL⇢ .Rd / d ⌧ < 1: 2

0

For p D 2; estimate (33) is established in the same way. By using estimates (26), (27) (30), (31), and (33), for u 2 Bp;T;⇢ ; we get p k‰ukBp;T;⇢

� �p � 4 � X � � � D� Ij .t /� � �j D0 �

Bp;T;⇢

i.e., the operator ‰ maps the space Bp;T;⇢ into itself.

p�1

5

4 X

j D0

p

kIj .t /kBp;T;⇢ < 1;

(33)

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

327

We now establish the property of contraction. Note that, for any fu; vg ⇢ Bp;t;⇢ ; we have

k‰u �

p ‰vkBp;T;⇢

� �p �X � 4 � � � D � .Ij .s/.u/ � Ij .s/.v//� � �j D1 �

p�1

4

4 X

j D1

Bp;T;⇢

p

kIj .s/.u/ � Ij .s/.v/kBp;T;⇢ :

Thus, by virtue of inequalities (27), (30), (31), and (33), we can estimate the quantities p

kIj .s/.u/ � Ij .s/.v/kBp;T;⇢ ;

j 2 f1; : : : ; 4g;

as follows: p

kI1 .s/.u/ � I1 .s/.v/kBp;t;⇢ � �p �Z � � � � � D sup E � .b.s; x; u.˛.s/; ⇠/; ⇠/ � b.s; x; v.˛.s/; ⇠/; ⇠//d ⇠ � � 0st � �Rd � ⇢

L2 .Rd /

0

Z

B  sup @ 0st

Rd

0

B D sup @ 0st

Z

Rd

0

Z

B @

Rd

0 B @

Z

1

1 p2

1

1 p2

l 2 .s; x; ⇠/ C C d ⇠ A ⇢.x/dx A ⇢.⇠/ l 2 .s; x; ⇠/ ⇢.⇠/

Rd

p

sup ku.s/ � v.s/kL⇢ .Rd / 2

0st

C C p d ⇠ A ⇢.x/dx A ku � vkBp;t;⇢ ;

p

kI2 .s/.u/ � I2 .s/.v/kBp;t;⇢ � 0 �Zs Z � � B D sup E � @Åx K.s � ⌧; x � ⇠/ 0st � �0 Rd 0

B ⇥@

Z

Rd

�p � � C C � .b.⌧; ⇠; u.˛.⌧ /; ⇣/; ⇣/ � b.⌧; ⇠; v.˛.⌧ /; ⇣/; ⇣//d ⇣ A d ⇠ A d ⌧ � � � ⇢ 1

L2 .Rd /

0

B  C t p sup @ 0⌧ t

1

Z Z

Rd Rd

2 .⌧; x; ⇣/

⇢.⇣/

1 p2

C d ⇣dx A

p

sup ku.⌧ / � v.⌧/kL⇢ .Rd /

0⌧ t

2

A. N. S TANZHITSKII AND A. O. T SUKANOVA

328

0

B D C t p sup @ 0⌧ t

Z Z

2 .⌧; x; ⇣/

⇢.⇣/

Rd Rd

1 p2

C p d ⇣dx A ku � vkBp;t;⇢ ;

p

kI3 .s/.u/ � I3 .s/.v/kBp;t;⇢ � �2 �Zs Z � � � � � D sup E � K.s � ⌧; x � ⇠/.f .⌧; u.˛.⌧//; ⇠/ � f .⌧; v.˛.⌧ //; ⇠//d ⇠d ⌧ � � � 0st � � ⇢ 0 Rd

L2 .Rd /

p

p

p

p

 cC⇢2 .T /Lp t p sup ku.⌧ / � v.⌧/kL⇢ .Rd / D cC⇢2 .T /Lp t p ku � vkBp;t;⇢ ; 0⌧ t

2

p

kI4 .s/.u/ � I4 .s/.v/kBp;t;⇢ � Zs 1 � Xp � D sup E� �n 0st � 0 nD1

Z

K.s � ⌧; x � ⇠/

Rd

�p � � ⇥ .�.⌧; u.˛.⌧//; ⇠/ � �.⌧; v.˛.⌧ //; ⇠//en .⇠/d ⇠ dˇn .⌧ /� � ⇢



c

Dc

1 X

�n

nD1 1 X

nD1 1 X

nD1

! p2

!

!

0

p 2

C⇢ .T /Lp Cp @ p

Zt 0

L2 .Rd /

1 p2

2 d⌧A ku.˛.⌧// � v.˛.⌧ //kL ⇢ .Rd / 2

p

p

�n C⇢2 .T /Lp Cp t 2 sup ku.⌧ / � v.⌧/kL⇢ .Rd / !

0⌧ t

p

p

2

p

�n C⇢2 .T /Lp Cp t 2 ku � vkBp;t;⇢ :

For any fu; vg ⇢ Bp;t;⇢ ; this yields 0

0

B B p k‰u � ‰vkBp;t;⇢  4p�1 B @ sup @ 0st

0

Z

Rd

0

B C C t p sup @ 0⌧ t

B @

Z

l 2 .s; x; ⇠/ ⇢.⇠/

Rd

Z Z

Rd Rd

C C d ⇠ A ⇢.x/dx A

2 .⌧; x; ⇣/

⇢.⇣/

1 p2

1

1 p2

C d ⇣dx A

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM p 2

Cc C⇢ .T /Lp t p C c

1 X

!

329

p 2

�n C⇢ .T /Lp Cp t

nD1

p 2

!

p

ku � vkBp;t;⇢

p

D �.t /ku � vkBp;t;⇢ : By virtue of (25), the first term in � is smaller than 1: Thus, choosing sufficiently small 0  t1  T; we conclude that 0  �.t1 / < 1: This means that the operator ‰ given in the Banach space Bp;t1 ;⇢ is contracting and, hence, possesses a unique fixed point, which is a solution u 2 Bp;t1 ;⇢ of the equation ‰u D u: This procedure can be repeated finitely many times on the other sufficiently small segments Œt1 ; t2 ç; Œt2 ; t3 ç; : : : ; Œtn�2 ; tn�1 ç; Œtn�1 ; T ç that form Œ0; T ç: As a result, we obtain the required solution as the union of solutions on these segments. This proves the theorem. Example . We now consider the Cauchy problem 0

d @u.t; x/ C

Z p

¶

c⇢.⇠/ exp �t � j⇠j � 2x

R

· 2

1

sin u.t � h; ⇠/d ⇠ A

D .Åx u.t; x/ C f .t; x/ cos u.t � h; x// dt C g.t; x/ sin u.t � h; x/d W .t; x/; 0 < t  T; u.t; x/ D �.t; x/;

x 2 R;

h  t  0;

x 2 R;

h > 0;

where p p

16 0
p

· ¶ c⇢.⇠/ exp �t � j⇠j � 2x 2 j sin u � sin vj

p · ¶ c⇢.⇠/ exp �t � j⇠j � 2x 2 ju � vj D l.t; x; ⇠/ju � vj;

l.t; x; ⇠/ D

p

c⇢.⇠/ expf�t � j⇠j � 2x 2 g:

(34)

A. N. S TANZHITSKII AND A. O. T SUKANOVA

330

We now check conditions (17) and (18) for l:

sup

0t T

Z

R

0 @

Z

l 2 .t; x; ⇠/ ⇢.⇠/

R

D sup

0t T

R

0 @

Z

d ⇠ A ⇢.x/dx c⇢.⇠/ expf�2t � 2j⇠j � ⇢.⇠/

4x 2 g

R

1

d ⇠ A ⇢.x/dx

1 ! 0Z Z A @ sup expf�2t g expf�2j⇠jgd ⇠ expf�4x 2 g⇢.x/dx

Dc

c

Z

1

0t T

R

R

p c ⇡ expf�4x gdx D < 1; 2

Z

2

R

Z v uZ 2 u l .t; x; ⇠/ t d ⇠ dx sup ⇢.⇠/ 0t T R

R

Z v uZ u c⇢.⇠/ expf�2t � 2j⇠j � 4x 2 g t D sup d ⇠ dx ⇢.⇠/ 0t T R

D

D

p

c

r

R

1v ! 0Z uZ · ¶ u 2 A @ sup expf�t g exp �2x dx t expf�2j⇠jgd ⇠

0t T

R

R

c⇡ < 1; 2

i.e., conditions (17) and (18) are satisfied. Further, we establish the validity of conditions (19)–(25) by using the following transformations: ·p · ¶ ¶ c⇢.⇠/ exp �t � j⇠j � x 2 j sin uj j@x b.t; x; u; ⇠/j D 4x exp �x 2 r

2 D

· ¶ 2p c⇢.⇠/ exp �t � j⇠j � x 2 .1 C juj/ e

1 .t; x; ⇠/.1

C juj/;

·p · ¶ ¶ c⇢.⇠/ exp �t � j⇠j � x 2 j sin uj j@2x b.t; x; u; ⇠/j  4.1 C 4x 2 / exp �x 2

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

 D

331

· ¶ 20 p c⇢.⇠/ exp �t � j⇠j � x 2 .1 C juj/ e 2 .t; x; ⇠/.1

C juj/;

ˇp ˇ ¶ · j@2x b.t; x; u; ⇠/ � @2x b.t; x; v; ⇠/j D 4 ˇ1 � 4x 2 ˇ c⇢.⇠/ exp �t � j⇠j � 2x 2 j sin u � sin vj 

D

20 p c⇢.⇠/ expf�t � j⇠j � x 2 gju � vj e 2 .t; x; ⇠/ju

� vj;

i.e., we arrive at conditions (19) and (20), where .t; x; ⇠/ D maxf

1 .t; x; ⇠/;

2 .t; x; ⇠/g

D

20 p c⇢.⇠/ expf�t � j⇠j � x 2 g: e

We now check the validity of conditions (21), (22), and (24) for

sup

0t T

Z

R

0 @

D

12

Z

.t; x; ⇠/d ⇠ A dx

R

400 e2

400c D 2 e

sup

0t T

R

R

12 ! 0Z Z p · ¶ exp �2x 2 dx sup expf�2t g @ ⇢.⇠/ expf�j⇠jgd ⇠ A

0t T

r

400c  2 e

12 Z p · ¶ @ c⇢.⇠/ exp �t � j⇠j � x 2 d ⇠ A dx 0

Z

sup

0t T

400c D 2 e

Z Z

:

r

0

⇡@ 2

Z

R

⇡ 2

Z

R

1

⇢.⇠/d ⇠ A

Z

expf�2j⇠jgd ⇠

R

⇢.⇠/d ⇠ < 1;

R

2 .t; x; ⇠/

⇢.⇠/

d ⇠dx

R R

400 D 2 e

sup

0t T

Z Z

R R

· ¶ c⇢.⇠/ exp �2t � 2j⇠j � 2x 2 d ⇠ dx ⇢.⇠/

R

A. N. S TANZHITSKII AND A. O. T SUKANOVA

332

D

1 ! 0Z Z · ¶ A @ sup expf�2t g expf�2j⇠jgd ⇠ exp �2x 2 dx

400c e2

0t T

r

400c D 2 e

R

R

⇡ < 1; 2

i.e., we arrive at conditions (21) and (22). We fix a point x0 2 R and prove that there exists a function '.t; ⇠; x0 ; ı/; ı 2 RC ; from condition (24) satisfying condition (23): .t; x0 ; ⇠/j 

j .t; x; ⇠/ �

20 p c⇢.⇠/ expf�t � j⇠jg.ı C 2jx0 j/jx � x0 j e

D '.t; ⇠; x0 ; ı/jx � x0 j; i.e., the function

satisfies condition (24) with the function '.t; ⇠; x0 ; ı/ D

20 p c⇢.⇠/ expf�t � j⇠jg.ı C 2jx0 j/: e

We now verify condition (23) for this function: sup

0t T

Z

' 2 .t; ⇠; x0 ; ı/ d⇠ ⇢.⇠/

R

400 D 2 .ı C 2jx0 j/2 sup e 0t T 400c D 2 .ı C 2jx0 j/2 e

D

Z

c⇢.⇠/ expf�2t � 2j⇠jg d⇠ ⇢.⇠/

R

sup expf�2t g

0t T

!Z

expf�2j⇠jgd ⇠

R

400c .ı C 2jx0 j/2 < 1: 2 e

Condition (25) takes the form 0

sup @

0t T

Z

R

0 @

Z

R

l 2 .t; x; ⇠/ ⇢.⇠/

1

1 p2

d ⇠ A ⇢.x/dx A

p p ! p2 ✓ p ◆ p2 p c ⇡ 16 ⇡ D < p 2 8 ⇡ 2

E XISTENCE AND U NIQUENESS OF THE S OLUTION TO THE C AUCHY P ROBLEM

p p

16 16

D

333

! p2

D

1 4p�1

;

which means that the conditions of Theorem 1 are satisfied for problem (34). 6. Corollary of the Theorem In a special case of problem (1), i.e., for the initial-value problem 0

B d @u.t; x/ C

Z

Rd

1

C b.t; x; ⇠/u.t � h; ⇠/d ⇠ A

D .Åx u.t; x/ C f .t; u.t � h/; x// dt C � .t; u.t � h/; x/d W .t; x/; u.t; x/ D �.t; x/;

�h  t  0;

x 2 Rd ;

0 < t  T;

x 2 Rd ;

(35)

h > 0;

the following theorem is true: Theorem 2. Suppose that the following conditions are satisfied: (i) ff; �gW Œ0; T ç ⇥ R ⇥ Rd ! R are the functions from assertion (i) in Theorem 1; (ii) bW Œ0; T ç ⇥ Rd ⇥ Rd ! R is a measurable function such that sup

0t T

Z

Rd

and

0 B @

Z

Rd

1

b 2 .t; x; ⇠/ C d ⇠ A ⇢.x/dx < 1 ⇢.⇠/

Z v u Z b 2 .t; x; ⇠/ u sup d ⇠dx < 1I t ⇢.⇠/ 0t T Rd

Rd

(iii) for any x 2 Rd ; there exist the derivatives @xi b and @xi xj b; fi; j g ⇢ f1; : : : ; d gI moreover, the gradient rx b; and the matrix Dx2 b satisfy the conditions � � jrx b.t; x; ⇠/j C �Dx2 b.t; x; ⇠/� 

where the function

.t; x; ⇠/;

0  t  T;

satisfies the conditions of assertion (iii) in Theorem 1.

Under these conditions, if 0

B sup @

0t T

Z

Rd

0 B @

Z

Rd

1

1 p2

b 2 .t; x; ⇠/ C C d ⇠ A ⇢.x/dx A ⇢.⇠/

<

1 ; 4p�1

fx; ⇠g ⇢ Rd ;

A. N. S TANZHITSKII AND A. O. T SUKANOVA

334

then problem (35) possesses a unique mild solution u 2 Bp;T;⇢ on 0  t  T: REFERENCES 1. Y. El Boukfaoui and M. Erraoui, “Remarks on the existence and approximation for semilinear stochastic differential equations in Hilbert spaces,” Stochast. Anal. Appl., No. 20, 495–518 (2002). 2. L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI (1998). 3. T. E. Govindan, “Autonomous semilinear stochastic Volterra integrodifferential equations in Hilbert spaces,” Dynam. Syst. Appl., No. 2, 51–74 (1994). 4. T. E. Govindan, “Stability of mild solutions of stochastic evolution equations with variable delay,” Stochast. Anal. Appl., 5, No. 2, 1059–1077 (2002). 5. V. Kolmanovskii, N. Koroleva, T. Maizenberg, etc., “Neutral stochastic differential delay equations with Markovian switching,” Stochast. Anal. Appl., 4, No. 21, 819–847 (2002). 6. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI (1968). 7. K. Liu and X. Xia, “On the exponential stability in mean square of neutral stochastic functional differential equations,” Syst. Control Lett., 4, No. 37, 207–215 (1999). 8. N. I. Mahmudov, “Existence and uniqueness results for neutral FSDEs in Hilbert spaces,” Stochast. Anal. Appl., 1, No. 24, 79–97 (2006). 9. A. M. Samoilenko, N. I. Mahmudov, and A. N. Stanzhitskii, “Existence, uniqueness, and controllability results for neutral FSDEs in Hilbert spaces,” Dynam. Syst. Appl., 17, 53–70 (2008). 10. X. Mao, “Asymptotic properties of neutral stochastic differential delay equations,” Stochast. Rep., 3–4, No. 68, 273–290 (2000). 11. X. Mao and X. X. Liao, “Exponential stability in mean square of neutral stochastic differential difference equations,”Dynam. Contin. Discrete Impuls. Syst., 4, No. 6, 569–586 (1999). 12. X. Mao, A. Rodkina, and N. Koroleva, “Razumikhin-type theorems for neutral stochastic functional-differential equations,” Funct. Different. Equat., 1–2, No. 5, 195–211 (1998). 13. X. Mao, “Razumikhin-type theorems on exponential stability of neutral stochastic functional-differential equations,” SIAM J. Math. Anal., 2, No. 28, 389–401 (1997). 14. M. McKibben, “Second-order neutral stochastic evolution equations with heredity,” Appl. Math. Stochast. Anal., 2, No. 2004, 177–192 (2004). 15. A. Sigurd and R. Manthey, “Invariant measures for stochastic heat equations with unbounded coefficients,” Stochast. Proc. Appl., 103, 237–256 (2003). 16. G. Tessitore and J. Zabczyk, “Invariant measures for stochastic heat equations,” Probab. Math. Statist., 18, 271–287 (1998). 17. J. Zabczyk and G. Da Prato, Ergodicity for Infinite-Dimensional Systems, Cambridge Univ. Press, Cambridge (1996). 18. J. Zabczyk and G. Da Prato, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge (1992).