Linear and nonlinear abstract differential equations of

Linear and nonlinear abstract di¤erential equations of high order

Veli B. Shakhmurov Okan University, Department of Mechanical Engineering, Ak…rat, Tuzla 34959 Istanbul, Turkey, E-mail: [email protected] Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences MSC 2010: 35xx, 47Fxx, 47Hxx, 35Pxx Abstract The nonlocal boundary value problems for linear and nonlinear degenerate abstract di¤erential equations of arbitrary order are studied. The equations have the variable coe¢ cients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding di¤erential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established. In application, the separability and spectral properties of nonlocal boundary value problem for the system of degenerate di¤erential equations of in…nite order is derived Key Words: abstract di¤erential equations, semigroups of operators, abstract function spaces, nonlocal boundary value problems, spectral properties of di¤erential operators 1. Introduction, notations and background Boundary value problems (BVPs) for abstract di¤erential equations (ADEs) have been studied extensively for example, in [3], [5], [8-23], [27]. The maximal regularity properties of ADEs have been investigated in [1, 2] ; [8-9] ; [16-23] ; for instance. Degenerate ADEs were studied in [11], [20-22] and elsewhere. The main aim of this paper is to discuss the BVP for the following degenerate linear ADE of arbitrary order Lu = "a (x) u

[m]

(x) + A (x) u (x) +

m X1

i

" m Ai (x) u[i] (x) = f (x) ;

i=1

and the nonlinear degenerate equation b (x) u[m] (x) + B x; u; u[1] ; :::; u[m = F x; u; u[1] ; :::; u[m 1

1]

1]

u (x)

; x 2 (0; 1) ;

(0.1)

where " is a small parameter, m 2 is an integer number, a (:), b (:) are complexvalued functions, A (:), Ai (:) are linear and B (:) ; F (:) are nonlinear operators function in a Banach space E and Dx[i] u = u[i] (x) = x

d dx

i

u (x) ;

0.

The uniform Lp –maximal regularity properties, resolvent estimates and Fredholm property of the linear problem is obtained. Especially, we prove that the corresponding realization operator is R-positive and also a generator of the analytic semigroup. Note that the principal part of the corresponding di¤erential operator is non self-adjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent, Fredholmness, discreetness of the spectrum and completeness of root elements of this operator is established. Then, by using the maximal regularity properties of linear problem (0:1), the existence and uniqueness of maximal regular solution of nonlinear problem is derived. In application, the BVP for system of parameter dependent degenerate di¤erential equations of arbitrary order is studied. One of the important characteristics of these problems are the following: 1) boundary conditions are nonlocal; 2) the principal part of the problems are non self-adjoint; 3) the equations possesses the variable coe¢ cients with small parameter. Note that the maximal regularity properties of ADEs in E-valued function spaces were treated in [2], [4], [9], [16], [21-23], [26] and elsewhere. Nonlinear BVP for DOEs were studied in [1], [2], [14], [17], [23] ; for instance. Let = (x) be a positive measurable function on a domain Rn : Lp; ( ; E) denotes the space of strongly measurable E-valued functions that are de…ned on with the norm kf kp; = kf kLp;

( ;E)

=

Z

p

kf (x)kE (x) dx

1 p

;1

p < 1:

For (x) 1 the space Lp; ( ; E) will be denoted by Lp = Lp ( ; E) : The weight funct¬on is sa¬d to be the Muckenhopt type, i.e., 2 Ap ; 1 < p < 1 if there is a positive constant C such that 0 10 1p 1 Z Z 1 1 1 @ p 1 (x) dxA (x) dxA @ C; jQj jQj Q

Q

for all cubes Q Rn : The Banach space E is called an U M D-space if the Hilbert operator Z f (y) (Hf ) (x) = lim dy "!0 x y jx yj>"

is bounded in Lp (R; E) ; p 2 (1; 1) (see. e.g. [6, 7]). U M D spaces include e.g. Lp , lp spaces and Lorentz spaces Lpq ; p, q 2 (1; 1). 2

Let C denote the set of the complex numbers and S' = f ;

2 C, jarg j

'g [ f0g ; 0

'< :

A linear operator A is said to be '-positive in a Banach space E with bound 1 1 M (1 + j j) for any M > 0 if D (A) is dense in E and (A + I) L(E)

2 S' ; 0 ' < ; where I is the identity operator in E and L (E) denote the space of bounded linear operators in E: Sometimes A + I will be written as A + and will be denoted by A : It is known [24, §1.15.1] that a positive operator A has well-de…ned fractional powers A . De…nition 1.1: A Banach space E is said to be a space satisfying multiplier condition with respect to p 2 (1; 1) and weighted function , if for any 2 C (1) (( 1; 1) ; L (E)) the R-boundedness ( see e.g. [9, § 4.1] ) of the set n o j (j) ( ) : 2 ( 1; 1) n f0g ; j = 0; 1

implies that is a Fourier multiplier in Lp; (Rn ; E) : Note that if E is an UMD space, then this condition implies that is a Fourier multiplier in Lp (Rn ; E) (see for example [9, § 3.4] , [26]). Let E1 and E2 be two Banach spaces in a locally convex space. By (E1 ; E2 ) ;p , 0 < < 1; 1 p 1 we will denote the interpolation spaces obtained from fE1 ; E2 g by the K-method (see e.g. [15] ; [24, §1.3]). The operator A (x) is said to be '-positive in E uniformly with respect to x 2 G with bound M > 0 if D (A (x)) is independent of x, D (A (x)) is dense 1 M 2 S (') ; 0 ' < , where M is in E and (A (x) + ) 1+j j for all independent of x: The '-positive operator A (x) ; x 2 G is said to be uniformly R-positive in a Banach space E if there exists ' 2 [0 ; ) such that the set n o 1 A (x) (A (x) + I) : 2 S' is uniformly R-bounded, that is nh sup R A (x) (A (x) + I) x2G

where R fB ( ) ;

1

i

:

2 S'

o

M;

2 Gg denotes the R-bound (see e.g [9]) of the set fB ( ) ;

2 Gg

L (E) :

Assume E0 , E are two Banach spaces and E0 is continuously and densely embeds into E. Consider the E-valued weighted function space de…ned by n o m Wp; (a; b; E0 ; E) = fu 2 Lp; (a; b; E0 ) ; u(m) 2 Lp; (a; b; E) with the norm

kukWp;m

(a;b;E0 ;E)

= kukLp;

(a;b;E0 )

3

+ u(m)

Lp; (a;b;E)

< 1:

Consider now, the following E-valued weighted function space [m] [m] Wp; = Wp; (a; b; E0 ; E) = fu 2 Lp (a; b; E0 ) ; u[m] 2 Lp (a; b; E) ;

kukW [m] = kukLp (a;b;E0 ) + u[m] p;

Lp (a;b;E)



m : 1+m

(3.4)

Then from (3:3) and (3:4) by virtue of Theorem 1.4 we obtain the assertion (b). Let V" denote the operator in Lp (0; 1; E) generated by BVP (2:1) (2:2) for = 0. From theorems 2.1, 3.1 and Remark 2.1we obtain the following Conclusion 3.1. Let all conditions of Theorem 3.1 be satis…ed. Then: [m] (a) operator V" is Fredholm from Wp; (0; 1; E (A) ; E) into Lp (0; 1; E) ; 1 (b) the resolvent operator (V" + ) for 2 S (') satis…es the following uniform coercive sharp estimate m X i=0

i

1

"m j j

i m

1

D[i] (V" + )

L(Lp (0;1;E))

+ A (V" + )

1 L(Lp (0;1;E))

C;

(c) the system of root functions of the di¤erential operator G" is complete in Lp (0; 1; E) : 4. Degenerate ADEs in moving domains Consider …rst, the inhomogeneous BVP for DOE with constant coe¢ cients on the moving domain (0; b (s)) au[m] (x) + (A + ) u (x) = f (x) ; k h i X [i] [i] ki u (0) + ki u (b (s)) = fk ; k = 1; 2; :::; m;

(4.1)

i=0

i

d u; a; ki ; ki are complex numbers; where u[i] = x dx parameter, k 2 f0; m 1g and A is a linear operator in E and i

=

i 1 + 2 mp (1

)

is a complex

:

From Theorem 1.3 we obtain Conclusion 4.1. Let the conditions of Theorem 1.3 are satis…ed for " = 1 and b = b (s) is a continuous function on [c; d]. Then the problem (4:1) for [m] f 2 Lp (0; b; E) and fk 2 Ek has a unique solution u 2 Wp; (0; b; E (A) ; E) : Moreover, for jarg j ' and su¢ ciently large j j the following uniform coercive estimate holds m X i=0

1

j j

i m

u[i]

Lp (0;b;E)

+ kAukLp (0;b;E) 13

"

C kf kLp (0;b;E) +

m X

k=1

#

kfk kEk :

Consider the BVP (2:1)

(2:2) in the moving domain (0; b (s)), i.e.,

a (x) u[m] (x) + A (x) u (x) +

m X1

Ai (x) u[i] (x) = f (x) ;

(4.2)

i=1

k h X

ki u

[i]

i [i] u (b (s)) = 0; k = 1; 2; :::; m; ki

(0) +

i=0

where

u[i] =

x

i

d dx

u (x) ; 0

< 1:

From the Theorem 2.1 we obtain Conclusion 4.2. Suppose b = b (s) is a continuous function on [c; d]. Assume the conditions of Theorem 2.1 are satis…ed for a.e. s 2 [c; d]. Then prob[m] lem (4:2) has a unique solution u 2 Wp; (0; b; E (A) ; E) for f 2 Lp (0; b (s) ; E), p 2 (1; 1) and 2 S' with su¢ ciently large j j and the coercive uniform estimate holds m X i=0

i m

1

j j

u[i]

Lp (0;b;E)

+ kAukLp (0;b;E)

kf kLp (0;b;E) :

Proof. Really, under the substitution = xb 1 (s) the moving BVP (4:2) maps to the following BVP with parameter in …xed domain (0; 1)

b

m

(s) a ( ) u[m] ( ) + A ( ) u ( ) +

m X1

b

i m

(s) Ai ( ) u[i] ( ) = f ( ) ;

i=1

k X

b

i

(s)

i=0

h

ki u

[i]

(0) +

i [i] u (1) = 0; k = 1; 2; :::; m: ki

Then by virtue of Conclusion 2.2 we obtain the assertion. 5. Nonlocal BVP for nonlinear degenerate ADE Consider now the following nonlinear problem b (x) u[m] (x) + B x; u; u[1] ; :::; u[m = F x; u; u[1] ; :::; u[m k h X

i=0

ki u

[i]

(0) +

ki u

[i]

1]

1]

u (x)

; x 2 (0; a) ;

i (a) = 0; k = 1; 2; :::; m;

14

(5.1)

(5.2)

i

d where u[i] (x) = x dx u (x) ; k 2 f0; m 1g ; b (x) is a complex-valued function, ki ; ki are complex numbers. Assume

j

k

0k

j+

k

1k

6= 0:

Let ! 1 ; ! 2 ; :::; ! m be the roots of the characteristic equation b (x) ! m + 1 = 0; where ! k = ! k (x), k = 1; 2; :::; m: In this section, we will prove the existence and uniqueness of maximal regular solution of the nonlinear problem (5:1) (5:2). Let [m] X = Lp (0; a; E) ; Y = Wp; (0; a; E (A) ; E) ; Ei = (E (A) ; E)

i;p ;

i

=

1 + ip (1 mp (1

) )

; X0 =

m Y1

Ei :

i=0

Remark 5.1. By using Lions-Peetre trace theorem (see e.g [24, § 1.8.]) and the Remark 2.1, we obtain that the embedding Di Y 2 Ei is continuous and there is a constant C1 such that for w 2 Y; W = fwi g ; wi = Di w ( ) ; i = 0; 1; :::; m 1; kuk1;X0 =

m Y1

Di w

C([0;a];Ej )

i=0

= sup

m Y1

Di w (x)

x2[0;a] i=0

Ej

C1 kwkY .

Consider the linear problem, b (x) w[m] (x) + (A (x) + d) w (x) = f (x) ;

(5.3)

Lk w = 0; k = 1; 2; :::; m; where A (x) is a linear operator in a Banach space E for x 2 (0; a), Lk are boundary conditions de…ned by (5:2) and d > 0: Assume E is a Banach space satisfying the multiplier condition with respect 1 to p and weighted function x 1 ; 0 < 1 p ; p 2 (1; 1) and A (x) is uniformly R-positive in E. By virtue of Conclusion 4.2, there is a unique solution w 2 Y of the problem (5:3) for all f 2 X and for su¢ ciently large d > 0. Moreover, the following coercive estimate holds kwkY

C0 kf kX ;

where the constant C0 is independent of f 2 X and a 2 (0 a0 ] : For r > 0 we denote by Or the closed ball in X0 with center w0 = Di w (0) ; i = 0; 1; 2; :::; m 1 of radios r, i.e., Or = u 2 X0 , ku 15

w0 kX0

r :

Condition 5.1. Assume the following satis…ed: (1) m 2; b (x) is a continuous function on [0; a] ; Re ! k 6= 0 and !k 2 S ('), for a.e. x 2 [0; a] ; b (0) = b (a) and 6= 0; (2) E is a Banach space satisfying the multiplier condition with respect to p and weighted function x 1 ; 0 < 1 p1 ; p 2 (1; 1); (3) F : [0; a] X0 ! E is a measurable function for each U = (u0 ; u1 :::; um 1 ) 2 X0 ; F (x; U ) is continuous as a function of x 2 [0; a] for all U 2 X0 and f (x) = F (x; 0) 2 X: Moreover, for each r > 0 there exists the positive functions hk (x) such that kF (x; U )kE F (x; U )

F x; U

h1 (x) kU kX0 ; E

h2 (x) U

U

X0

;

where hk 2 Lp (0; a) with khk kLp (0;a) < C0 1 , k = 1; 2; and U = fu0 ; u1 ; :::; um 1 g, U = fu0 ; u1 ; ::::um 1 g, ui ; ui 2 Ei and U; U 2 Or : (4) there exist i 2 Ei , such that the operator B (x; ) for = f i g is R-positive in E for ' 2 [0; ) uniformly in x 2 [0; a] ; B (x; ) B 1 x0 ; 2 C ([0; a] ; L (E)) for a.e. x; x0 2 [0; a] and for B (x; 0) = A (x) ; A (0) = A (a) ; (5) B (x; U ) for x 2 (0; a) is a uniform positive operator in a Banach space E and the operator B: (0; a) X0 ! L (E (A) ; E) is continuous. Moreover, for each r > 0 there is a positive constant L (r) such that B (x; U )

B x; U

E

L (r) U

U

X0

kA kE

for x 2 (0; a), U; U 2 Or and 2 D (B (x; U )) ; Theorem 5.1. Let the Condition 5.1 be satis…ed. Then there is a number a 2 (0 a0 ] such that problem (5:1) (5:2) has a unique solution belonging to space Y: Proof. We want to solve the problem (5:1) (5:2) locally by means of maximal regularity properties of the linear problem (5:3) via the contraction mapping theorem. For this purpose, let w be a solution of the linear problem (5:3): Consider a ball Br = f 2 Y; Lk (

w) = 0; k

wkY

rg :

Let w 2 Y be a solution of the problem (5:3) and W = w (0) ; w[1] (0) ; :::w[m For given

1]

(0) :

2 Br we will solve the following linear problem b (x) u[m] (x) + A (x) u (x) + du = F (x; ) + 16

[B (x; 0)

B (x; )] (x) ; Lk u = 0; k = 1; 2; :::; m;

(5.4)

where V =

;

[1]

; :::;

[m 1]

,

2 Y:

Consider the function (x) = F (x; ) + [B (x; 0)

B (x; )] (x) :

Let …rst, we show that 2 X and k kX C0 1 r for by Remark 5.1, 2 C ([0; a] ; X0 ). So, one has B (x; 0)

2 Y; k kY

r: Indeed,

B (x; ) 2 C ([0; a] ; L (E (A) ; E)) :

Hence, by assumption (3),

is measurable and L (r) k kX0 kA kX + h (t) k kX0 :

k kX

Then, by using the Remark 5.1 we obtain k kX

rL (r) k kX + r kh1 kLp

r2 L (r) + r kh1 kLp

r:

De…ne a map Q on Br by Q = u; where u is a solution of the problem (5:4) : We want to show that Q (Br ) Br and that Q is a contraction operator provided a is su¢ ciently small, and r is chosen properly. For this aim, by using maximal regularity properties of the problem (5:3) we have kQ

wkY = ku

wkY

k[B (x; 0) By assumption (3) for

C0 fkF (x; )

F (x; 0)kX +

B (x; V )] kX g :

2 Or we get

kF (x; )

F (x; 0)kX

kh2 kLp (0;a) k kX0 :

By assumptions (4), (5) and Remark 5.1, for 2 Or we have n k[B (x; 0) B (x; V )] kX sup k[B (x; 0) B (x; W )] kL(X0 ;X) + x2[0;a]

o B (x; V )] kL(X0 ;X) k kY h i L (r) kW kX0 kA kX + k wk1;X0 [k wkY + kwkY ] k[B (x; W )

rL (r)

kW kX0 k kY + C1 k

wkY +L (r) kwkY g :

Choosing r and a 2 (0 a0 ] so that kwkY < obtain from the above inequalities kQ

wkY

a;

by assumptions (3)-(5) we

r + r2 L (r) kW kX0 + r2 L (r) C1 + rL (r) kwkY < r: 17

That is the operator Q maps Br into itself, i.e., Q (Br ) Let u1 = Q (

1)

and u2 = Q (

2 ).

Br :

Then u1

u2 is a solution of the problem

b (x) u[m] (x) + A (x) u (x) + du = F (x; F (x; [B (x;

1) 1)

+ [B (x; B (x;

2)

2 )]

B (x; 0)] [ 1

1

(x)

1) 2

(x)]

(x) ; Lk u = 0; k = 1; 2; :::; m:

In a similar way, by using the assumption (5) we obtain ku1

u2 kY

C0 frL (r) k 1 2 kX + L (r) k 1 i o h C0 2rL (r) + kh2 kLp k kh2 kLp k 1 2 kY

2 kY 1

k

1 kX

2 kY

+

:

Thus Q is a strict contraction. Eventually, the contraction mapping principle implies a unique …xed point of Q in Br which is the unique strong solution u 2 Y 6. The system of degenerate di¤erential equation of in…nite order Consider the nonlocal BVP for the system of parameter dependent degenerate equation of in…nite order [m] "a (x) uj

(x) + (aj (x) + ) uj (x) +

m X1

i

[i]

" m aij (x) uj (x) = fj (x) ;

(6.1)

i=1

k X

i=0

"

i

h

[i] ki uj

(0) +

[i] ki uj

i (1) = 0; k = 1; 2; :::; m; x 2 (0; 1) ;

i

(6.2)

1 d u (x) ; 0 < 1; i = mi + mp ; k 2 f0; m 1g, j = where u[i] = x dx 1; 2; ::; N , N 2 N; ki ; ki are complex numbers, " is a positive and is a complex parameter; a (x), aij (x) and bij (x) are complex valued functions. We assume j k 0k j + 2: Let k 1k 6= 0 and m n lq (A) = u 2 lq ; kuklq (A) = kAuklq =

9 0 0 1 q1 q 1 q1 > N N = X X q @ @ A A = aj uj ; j=1 j=1 u = fuj g ; Au =

8 N