Sobolev type mathematical models with relatively positive operators in the sequence spaces

The Barenblatt–Zheltov–Kochina equation [1]

(X -A) ut = aAu + f

simulates the pressure dynamics of the fluid filtered in fractured porous media. Besides, the equation (1) simulates processes of moisture transfer in a soils [2] and processes of the solid-to-fluid thermal conductivity in the environment with two temperatures [3]. Note that the required function u = u(x, t) must be nonnegative, that is u > 0 by physical necessity. The Hoff equation [4] (X + A) ut = au + f

simulates the H-beam buckling under the influence of high temperatures. The case is also most interesting when the required function u = u ( x , t ) is nonnegative.

Consider both equations as special cases of Sobolev type mathematical model such as

Lu_t = Mu + f,                                    (3)

given in Sobolev sequence spaces m q

~ mq

, m e R, q e [ 1, +^ ) .

^u = { u k } : E ^X k^{2 u}k|^q <^

k = 0

Here L = L(Л) and M = M (Л) is polynomials with real coefficients, and their degrees satisfy the relation deg L > deg M ;

Л is transfer of the Laplace operator J to spaces I m , а { X k :X_k e R ₊ } is monotonically increasing sequence such as lim X_k= + ^ .

k -^^

The peculiarities of our approach will be, firstly, the active use of the theory of bounded operators and the degenerate holomorphic groups of operators generated by them [5, ch. 3]. Secondly, we apply the theory of positive groups of operators, defined on Banach lattices [6, ch. 2 and 3], to lay the foundations of the theory of positive degenerate holomorphic groups of operators whose phase spaces are Banach lattices. Thirdly, we consider the concrete mathematical model (3) in Sobolev sequence spaces l m , m e R, q e [1, +~ ), which can be interpreted as the space of Fourier coefficients of solutions

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of initial-boundary value problems for equations of the form (1) or (2). Let us note the difference between our approach and the ideas and methods proposed in [7].

The foundations of the theory of degenerate positive groups of operators theory are laid in the first part of the article, which are generated by relatively positively bounded operators. The degenerate positive holomorphic groups of operators obtained are applied to the study of the Cauchy problem solvability for the homogeneous (that is f ( t ) ≡ 0) abstract equation (3). The initial value is taken from the phase space of such an equation. In the second part, the solvability of the Showalter–Sidorov problem [8] for the abstract nonhomogeneous equation (3) was studied. Sufficient conditions are obtained for the existence of a positive solution of this problem. Abstract results are applied to a mathematical model of the form (3), where L = L ( Л ) and M = M ( Л ) are polynomials with real coefficients. It is noted that the Barenblatt-Zheltov-Kochina equation for Xa e R ₊ satisfies the sufficient conditions found, and therefore the initial-boundary value problem can have non-negative solutions. The final part of the article outlines directions for further possible research. The list of literature does not pretend to be complete and reflects only the tastes and preferences of the authors.

• 1.    Degenerate positive holomorphic groups of operators

Let U and F be Banach spaces, operators L e L ( U;F ) (i.e. linear and continuous), M e Cl ( U;F ) (i.e. linear, closed and densely defined). Sets p^L ( M ) = { ц e C : ( цL - M ) 1 e L ( F;U ) } and о ^L ( M ) = C \ p ^L ( M ) are called resolvent set and L - spectrum of operator M respectively. Operator M is ( L , a ) - bounded if

3 a e R ₊ V ц e C ( | ц | > a ) ^ ( ^ e p^L ( M )).

If operator M is ( L , 0) -bounded, then operators P, Q are the projectors

P =4-№( M ) d“ e L ( U ) , Q =^-U L ( M ) dц e L ( F ) . 2 πi                           2 πi

Y

Y

Here R L ( M ) = ( цL - M ) 1 L is called a right resolvent , and L ^ (M ) = L ( ц L - M )

¹ is is called a left

resolvent of operator M ; contour у = { ц e C : | ц | = r>a } . Here and below, loop integrals are understood in the sense of Riemann. We consider subspaces U ⁰ = ker P , U ¹ = im P, F ⁰ = ker Q, F ¹ = im Q ; and denote operator of the contraction L ( M ) on U k ( U k n dom M ) by L k ( M k ), k = 0,1.

Theorem 1.1. Let operator M be ( L , a ) -bounded. Then

• (i)    operators L k e L ( lJ ^k ;F ^k ) , k = 0,1 ; and there exist the operator L - ¹ e L ( F ¹ ;U ¹ ) ;

• (ii)    operators M k e Cl ( U k ;F ^k ) , k = 0,1 ; and there exist the operator M ₀ ¹ e L ( F ⁰ ;U ⁰ ) .

Let operator Mbe (L,o) -bounded, construct the operator H = M01L0 e L(U0). Operator M is called (L, p)-bounded, p e N, ((L,0) -bounded) if Hp ^ O, and Hp+1 = O (H = O). Let operator M be (L,p) -bounded, pe {0} и N , we consider the equation

Lu ɺ = Mu.                                         (5)

Vector function u = u (t), t e R , is solution of equation (5) if it satisfies this equation. Decision u = u (t) is called solution of the Cauchy problem u (0) = uо ,                                            (6)

if it satisfies condition (6) at some u ₀ e U . The set P c U is phase space of equation (5) if its any solution u ( t ) e P at each t e R ; and for any u ₀ e P there exists a unique solution u e C ¹ (R ;U ) of problem (6) for equation (5). Finally, we introduce a degenerate (if ker L ^ { 0 } ) holomorphic (in the whole plane ℂ ) group of operators

U = — [R R M ) e^du , t e C .

2πi    ^µ

Y notice, that U0 = P , where ker P ^ ker L .

Theorem 1.2. Let operator M be ( L , p ) - bounded , p e { 0 } u N . Then

• (i)    any solution u e C ¹ (R ;U ) of equation (5) has the form u ( t ) = U t u ₀ , t e R , and some u ₀ e U;

• (ii)    the phase space of equation (5) is subspace U ¹ .

Thus, under the conditions of the theorem 1.2 L -resolvent ( цк - M ) ¹ of operator M in the ring µ > a decomposes into a Laurent series

^                 p

( ^ L - M ) ^{- 1} = £ ц ^k S ^{k - 1} L - ¹ Q - £ ^H^k M о ¹ ( I - Q ) , k = 1                      k = 0

where operators S = L - ¹ M 1 e L ( U 1 ), H = M ₀ ¹ L ₀ e L ( U ⁰). Hence the resolving degenerate group U t of equation (5) is as follows

Ut = (I -Q) + eStQ, where e8 = 2171 (* I - S) e*"* = £ Sk 2n i у                    k=0 k• is the group of operators of equation (5), given on the phase space U1 .

Next, we give an order relation “ >  ”, compatible with both vector and metric structures, to U ¹ . In other words, we assume that ( U ¹ ; > ) is a Banach lattice. Recall those properties of Banach lattices, which will prove useful to us in the future. An arbitrary set X is called ordered if on X X X there is the relation of order >  , which satisfies the following axioms:

• (io)    x >  x for each x e X ;

• (iio)    ( x >  y ) a ( y >  x ) ^ ( x = y ) for any x, y e X ;

• (iiio)    ( x >  y ) a ( y >  z ) ^ ( x >  z ) for any x, y,z e X.

An ordered vector space X is called Riesz space if in addition, the following axioms are satisfied:

(ivo) ( x >  y ) ^ ( x + z >  y + z ) for all x, y,z e X .

(vo) ( x >  y ) ^ (a x >  a y ) for all x, y e X and each a e { 0 } u R ₊ .

The Riesz space X is called functional Riesz space if u v v, u a v e X for any u, v e X . Here ( u v v )( x ) = max ( u ( x ) , v ( x ) ) , ( u a v ) ( x ) = min { u ( x ), v ( x ) } .

The spaces C ( Q ), C ( Q ), where Qc R ⁿ is a domain, and spaces l_q , where q e [1, +~ ] are the classical functional Riesz spaces examples. In these examples u v v, u a v are defined pointwise, but if measure is given on Q , and It is possible to define these elements almost everywhere, then the Lebesgue spaces L_q ( Q ) , q e [1, +^ ] can be assigned to the to the functional Riesz spaces.

In the Riesz function space, the following elements can be defined u ₊= max { u ,0 } , and u - = min { - u ,0 } , so that u = u ₊- u - , and there is another element | u| = u₊+u - If norm ||-|| ^ is given on the Riesz functional space X and satisfies the axiom

(vio) (| u\ > |v |) ^(|| u| X >|| v| X ) for all u, v e X , then we call the Riesz function space X normed Riesz function space . A complete normed functional space is called a Banach lattice. Spaces C ( Q ) , C ( Q ) and L q ( Q ) with the qualifications specified above, as well as space l_q , where domain Q c R n , q e [1, +~ ] are examples of Banach lattices.

Further, let X is vector space. Convex set C c X we call a cone if

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(ic) C + C c C ;

(iic) aC c C for any a е { 0 } u R ₊ ;

(iiic) C n ( - C ) = { 0 } .

The cone C is called generative if

(ivc) C - C = X .

Now let X is Riesz space. We construct the set

X+ ={ x е X: x > 0}.

Proposal 1.1. Let X be a vector space, C c X is generative cone. Then X is Riesz space, where relative >  is given by

(x > y) ^ (x - y е C).

Proposal 1.2. Let X be Riesz space, then X₊ is generative cone.

Let X be Banach lattice with generative cone X₊ . Linear bounded operator A е L(X) is positive if Au >  0 for all u e X₊ . Holomorphic group of operators X i = { X t : X t е L ( X ) for all t е r } is called positive if X t u >  0 for all u е X₊ and t е R .

Proposal 1.3. Holomorphic group X ⁱ is called exactly positive when its generator is positive '

A= X ^t .

• V      i t =0

• 2.    Mathematical model in sequence spaces

Finally, let us return to the abstract problem (5), (6). We will be interested in its positive solution u = u ( t ), i.e. such that u ( t ) >  0 for all t е R . Therefore, we consider the phase space of equation (5) U ¹ Banach lattice, generated by a cone U + . ( L , p )-bounded operator M is positive ( L , p ) -bounded , p е { 0 } u N if Su е U + for any u е U + . The degenerate holomorphic group U i е C ” (R; L ( U )), generated ( L , p ) -by positive operator M is called a degenerate positive holomorphic group .

Theorem 1.3. Let operator M is positive ( L,p ) -bounded, p е { 0 } u N . Then for any u ₀ е U + there is the unique positive solution u = u ( t ) , t е R , of problem (5), (6), and it has the form u ( t ) = S t u ₀ .

Let U и F be Banach spaces, operators L е L ( U;F ) , M е Cl ( U;F ) , and operator M is ( L , p )-bounded, p е { 0 } u N. Consider a linear inhomogeneous equation of Sobolev type

L u ɺ = Mu + f .

Vector function u е C ( [0 ,т ) ;U ) n C ¹ ( ( 0 , t ) ;U ) , т е R₊ , is called solution of equation (7) if it satisfies this equation for some f = f ( t ). The solution u = u ( t ) of equation (7) is called solution of the Showalter – Sidorov problem [9]

lim P ( u ( t ) - u ₀ ) = 0, t ^0+ '           '

if it also satisfies the initial condition (8). Here P:U ^ U ¹ along U ⁰ is projector. Further, let U be a Banach lattice generated by the cone U₊ . The solution u = u ( t ) of problem (7), (8) is positive if u ( t ) е U₊ for any t е [0, 7 ).

We will be interested in the conditions under which the solution u = u ( t ) of problem (7), (8) is positive. Let F be also be a Banach lattice generated by a cone F₊ . If operator M is ( L , p ) -bounded, p е { 0 } u N , then it is not difficult to show that the subspaces U k and F k , k = 0,1, are also Banach lattices generated by cones U + = U ^k n U₊ and F^ = F ^k n F₊ , k = 0,1, respectively. ( L,p )-bounded operator M is called strongly positive if

(ip) operator L ₀ : U + 0 ^ F , and operator L 1 : U + ^ F + ¹ is a toplinear isomorphism;

(iip) operator M 1 : U + n dom M ^ F 1 and operator M 0 : U + n dom M ^ F 0 , and M 0 ¹ ^ F ₊ ⁰ J c U + 0 .

It is easy to see that strongly positive ( L , p ) - bounded operator M is positive ( L , p ) - bounded , p e { 0 } u N . Let be f = ( I - Q ) f + Qf = f ⁰ + f 1 , where Q : F ^ F ¹ is projector along F ₀ .

Theorem 2.1. Let U be a Banach lattice and operator M is strongly positive (L, p) -bounded, p e{0}u N. Then for any vector functions     f: [0 ,t)^ F     such that f0 e cp+1(( о ,t ); F0),-f 0( k)(t )eF0, k = 0, p +1, t e (0,7), f1 e C ([0 ,t ); F+), and for any vector u0 e U, such that u 0 e U+ there exists the unique positive solution u = u (t), which also has the form u (t) = - f Hk M 01 f 0(k) (t) + Uu 0 + J Ut-7L- f1 (7) d7.

k = 0                                0

k

Here f0(k) (t) = —r f0(t), k = 0, p +1. Proof of the theorem 2.1 does not differ fundamentally from dtk the proof of the theorem 5.1.1 [5]. We check the positivity of the resulting solution for the reader. We also note that condition - f0( k)(t )e F+0, t e (0,7), k = 0, p +1, seems difficult, so here is an example: f0(t)=-eaf0 , where ae Ж +, J, e F+0 .

We consider Sobolev sequence spaces I'm , m e R, q e [1, +«). First of all, we note that these spaces are Banach spaces with the norm

Г «ml A q u m,

I 42 Ы к k=1 J

Then pay attention to dense and continuous investments lm '^ iq at m > n . (The proof of this fact is left to the reader). Finally, we set operator Лu = (2^), where u = (uk). We show that operator Лe L (lm+2; lm). Indeed, qq mq q

IIЛ u m,q

« ^-+q fV   l«kl’

= u . m + 2, q

k = 1

к               J

Let's construct operators L = L ( Л ) and M = M ( Л ), where L ( s ) and M ( s ) are polynomials with real (for simplicity) coefficients. If the condition (4) is satisfied, that operators L, M e L ( l m ^{+ deg L} ; l m ), m ^e R , q ^{e [1, +}« ). Indeed, || ^u||m ₊ deg L, q > H ^u|| _{m+ deg M} , _q , u ^e l m+ deg ^L , m ^e R , q ^{e [1, +}« ). ^Hence, by ^the continuity of the embedding l_q ^{m+ deg L} l_q ^{m+ deg M} follows the truth of what has been said.

Lemma 2.1. Let

• (i)    the condition (4) is satisfied;

• (ii)    polynomials L = L ( s ) and M = M ( s ) have only real roots and have no common roots.

Then operator M is ( L , 0) -bounded.

Before proceeding with the proof of this assertion, we make a number of remarks. At first, let operators L, M e L ( U;F ) , where U and F are Banach spaces. If there exists a vector ^ e U such that My = Lф , where the vector ф e ker L \ { 0 } , then it is called the adjoint vector of operator L . Secondly, operator A e L ( U;F ) is called Fredholm operator, if dim ker A = codim im A . Third, the proof of Lemma 2.1 will be based on the following assertion, which is a particular case of Theorem 4.6.1 [5].