Asymptotic Integration of Certain Differential Equations in Banach Space

In this work, we investigate the problem of constructing asymptotic representations for weak solutions of a certain class of linear differential equations in the Banach space as an independent variable tends to infinity. We consider the class of equations that represent a perturbation of a linear autonomous equation, in general, with an unbounded operator. The perturbation takes the form of a family of bounded operators that, in a sense, oscillatorally decreases at infinity. It is assumed that the unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence of a center-like manifold (a critical manifold) for the initial equation. This manifold is positively invariant with respect to the initial equation and attracts all trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional system of ordinary differential equations. The asymptotics of the fundamental matrix of this system can be constructed by using the method developed by P.N. Nesterov for asymptotic integration of systems with oscillatory decreasing coefficients. We illustrate the proposed technique by constructing the asymptotic representations for solutions of the perturbed heat equation.


FORMULATION OF THE PROBLEM
We study the equation (1) where is an element of a complex Banach space . Here, is a closed linear operator with the dense domain in ; this operator is a generator of a strongly continuous semigroup of bounded linear operators ( ). In this case, ( ) is a family of linear bounded operators acting from to ; here, (2) In (2), the family of linear bounded operators is such that the operator function is strongly measurable on any interval , , while oscillatory tends to zero when for . More accurately, the structure of this family of operators will be defined later. The family of linear bounded operators is also strongly measurable on any interval , ; in addition, there exists a function such that (3) for any . The feasibility of studying equations of form (1), where an operator function is understood as a certain parametric perturbation with a continuous spectrum, is noted, among other things, in the monograph [6, p. 230].
Everywhere in this work, the solution of Eq. (1) with the initial condition is understood in the weak sense (see [9]), more precisely, as a solution of the integral equation From the results of [9,10] (see also [8]) it follows that for any there exists an unambiguous continuous (on the interval ( )) weak solution of Eq. (1) with the initial condition , and this solution is described by formula (4). The question we are interested in is the asymptotic behavior of solutions of Eq. (4) as if the operator is subjected to the following additional conditions. Suppose (i) (5) where the finite-dimensional linear subspace is a linear span of generalized eigenvectors of the operator corresponding to the eigenvalues with the zero real part (with account of their multiplicities); (ii) the closed linear subspace is invariant under the semigroup and, in addition, for any , the following inequality holds: (6) where . Formulated conditions (i) and (ii) represent standard requirements of the centre manifold theory (see, e.g., [3,11,20]). We use the basic ideas of this theory, along with the averaging-method version proposed in [4] for constructing the asymptotics of weak solutions of Eq. (1) as .

CRITICAL MANIFOLD AND ITS PROPERTIES
We begin this section by clarifying the type of the operator function in (2). In accordance with [4,6], we assume that (7) Here, are operator functions for which it is assumed that (8) where are linear bounded operators that do not depend on and act from to . Finally, are scalar functions absolutely continuous on such that . w h e n ; . ; . the product for any set .
and is a certain norm in the space of row-vectors of length N.
2. The set for is described by the formula for all . Before we derive the system of equations describing the dynamics of solutions of Eq. (1) on the critical manifold , recall the following facts from the theory of conjugate operators in Banach spaces. Suppose is a space conjugate to (a space of linear bounded functionals assigned on ) and are brackets of duality between the spaces and such that (11) for any and . Here, if and , then is an mby-l matrix such that Suppose A' is a conjugate to linear closed operator acting from the space to . As is well known (see, e.g., [1,14]) from assumption (i) on the operator , the operator A' also has eigenvalues ; here, the dimensions of matching generalized eigenspaces of these operators that correspond to the same eigenvalues, coincide. Denote by a column vector of length consisting of linearly independent generalized eigenvectors of the operator A' that correspond to the eigenvalues . Suppose the column vector is chosen in such a way that (12) where is an N-by-N unit matrix (see [7,13]). Note in particular that one of the possible choices of the complementary subspace for in condition (i) is the choice of the set for this subspace.

t T t t u T t s G s u s ds
We construct a system that describes the dynamics of solutions of Eq. (1) on a critical manifold assuming the existence of this manifold for sufficiently large . Suppose is a projector on the subspace along . Note that is a bounded linear operator defined in the entire space (see, e.g., [14]). On account of condition (i) we conclude that if , then (13) where (14) and by we denote an identity operator. Note that since the subspaces and are invariant under the operator and the semigroup , for all and we have the equalities (15) Substituting (13) in Eq. (4) and taking into account (14) and (15), we obtain Since

t T t t u t T t s P G s u s ds
Pu t w t for certain , we have (19) In addition, from (i) it follows that ; hence, there exists an N-by-N matrix whose spectrum is the set such that (20) Note that if an element belongs to the domain of the operator , then from [18, p. 4

H t T t t H t W t t T t s P G s H s W s t ds
We assume that the domain of the operator is the Banach space of continuous in row vectors of length with the values from the space and the fixed initial condition , such that when . The norm in this space we define as follows: (29) From (6), for any we have the estimate Here and further the various constants, whose exact values are not important to us, we will denote by the same symbols. In addition, from (2), (3), (7), and (8) it follows that (31) Here, as and is a certain function from the class . Further steps in proving rest on the principle of contracting mappings. In an analogous way as it is done in proving Theorem 1 in [5] (see also [17,Theorem 4.3]), it is easy to show that the operator maps a certain closed ball of the space to itself and represents a contracting operator in this ball if is sufficiently large and the chosen initial condition is sufficiently small.
For approximately finding the row vector , which describes the critical manifold by formula (10), we act as follows. Substitute representation (24) in Eq. (1) and take into account (20) We shall try to satisfy Eq. (32) with the accuracy of terms such that . We put Here, the elements of the row vectors of length N, to be determined, belong to the space for all and represent trigonometric polynomials in the sense (34) where . In addition, a value of the integer is determined by property of the functions . Substitute expression (33) into Eq. (32) instead of and gather the terms with the same multipliers ( ). We obtain the following one-type equations for determining the row vectors Here we take into account formulas (2), (7), and (8), from which it follows, specifically, that in order to solve Eq. (35), it is necessary to determine, first, all row vectors with . In this case, is a certain known row vector, which from (8) and (34) has the form analogous to (34): where the semigroup of operators has the form of (39). Proof. The convergence of the integral in (43) follows from the inequality (40). In an analogous manner to that which is done in [18, p. 8, Theorem 3.1], it is possible to show that integral (43) belongs to the domain of the operator described by formula (38). In addition, the following equality holds:

Dw t PG t H t w t Hw t H D PG t H t w t A w t AH t w t G t H t w t
which is equivalent to (37).Hence, the constructed row vector defined by formula (33) satisfies the following equation: We present the vector function as , where ( ) is the Cauchy matrix of system (25) ( ), in which represents sum (45). We substitute (50) in (49), return to (47), and obtain (with allowance for (28)) the following representation for the operator : In the same way as is done in proving Theorem 2 in [5] (see also [17,Theorem 4.4]), it can be established that the operator maps a certain closed ball of the space to itself and represents â

Here, is chosen on account of inequality (6), is an arbitrary value, and ( ) is a certain solution of the system on critical manifold (25).
Proof. On account of (13) and (18) We can consider that is so small as we need. In fact, on account of the linearity of Eq. (4), we can always pass from the solution to considering the solution for any preassigned Consider the space whose elements are pairs . The functions and are continuous when . Consider that the initial element is fixed and belongs to the space . In addition, the following inequalities hold:

t T t t z t T t s P G s r s z s ds t t * @
where a certain constant and arbitrarily taken are used. The space becomes the Banach space if we introduce in it the norm by the rule Note that if system (56) and (58) has a solution , then Eq. (58) can be written in the following equivalent form. In this equation, direct the variable to infinity; take into account the right of inequalities in (59), along with the fact that all eigenvalues of the matrix have zero real parts. We obtain Using this ratio in (58), rewrite the last expression: (60) We present system of Eqs. (56) and (60) in the form of an operator equation in the space : where the operators : and : are defined by the right sides of Eqs. (56) and (60), respectively. It can be shown that the operator is contractive in the space if is sufficiently small, is sufficiently large, and the constant in (59) is suitably chosen. This is done in exactly the same way as in the proof of Theorem 3 in [5] (see also [17,Theorem 4.7]).

Suppose
are fundamental solutions of the system on critical manifold (25), while is an arbitrary weak solution of Eq. (1) (this solution is determined for ). Then from Theorem 3, the following asymptotic representation holds when : ( ( ) ( )) ( ( ) ( )) ( ( ( ) ( )) ( ( ) ( ))) z t r t z t r t z t r t z t r t where are arbitrary complex constants and is a certain real number. Thus, the question of constructing asymptotics for weak solutions of Eq. (1) is reduced to the problem of asymptotic integration of the N-dimensional system of ordinary differential Eqs. (25).
With allowance for Theorem 3 and formulas (33) and (34), which define the row vector , the system on the critical manifold has the following form: (63) In this system, N-by-N matrices represent matrices whose elements are trigonometric polynomials, i.e., matrices of type (8), where are certain constant matrices. In addition, is a certain matrix from the class . System (63) belongs to the class of systems with oscillatory decreasing coefficients. The asymptotic method for integrating systems of such a type is proposed in [4]. The essence of this method is to conduct some special substitutions, which diagonalize the main part of system (63). More precisely, with the help of such substitutions, system (63) is brought to the L-diagonal form. The asymptotics of fundamental solutions of L-diagonal systems can be built using the theorem of N. Levinson (see [2,12,16]). The detailed description of the method for asymptotic integration ofsystems having form (63) can be found in [4,5,17].

EXAMPLE
As a simple example that illustrates the use of the technique described above, consider the perturbed heat equation Here the function is considered in a bounded area of space with a smooth boundary . We denote by the derivative along the exterior normal to . The real-valued functions and are considered as belonging to the space , the parameters and are positive, and is the Laplace operator in components of the vector . The question of constructing the asymptotics of solutions of Eq. (64) with conditions (65) and (66) is discussed in [15]. Here we show another approach to this problem.
We consider initial boundary value problem (64)-(66) in the Hilbert space . As the domain of the operator we consider the set of twice continuously differentiable in functions that meet boundary condition (66) in the boundary . It is known (see, e.g., [19,20]) that the operator defined in such a way, allows in the space the closure in the form of the operator . In turn, the operator is a generator of a strongly continuous (even analytic) compact semigroup of the operators . Note that the point spectrum of the operator has the form For this reason, present the space as direct sum (5), taking for the space the eigenspace (of the operator ) that corresponds to the eigenvalue . Then . Because is a Hilbert space, and duality bracket (11) is a scalar product in the space , , Note that is a self-conjugate operator; therefore, . The elements and of the space we choose in such a way that normalization condition (12) Here the function for all belongs to the space ; in addition, for . By Theorem 2, for this function we have the following representation: Here the function , which belongs to the subspace , is such that when and . The function , which belongs in the variable to the subspace , represents an approximate solution of the equation (71) with an accuracy of terms such that . Here we also take into account the fact that for any the following equality holds: We seek the function as (73) where the functions in the variable belong to the subspace and represent trigonometric polynomials of the variable . We substitute expression (73) in Eq. (71) and gather the terms with ; in doing so, we omit terms such that . By using (67), we obtain the following equation for finding the function :

( ) ( ) ( ) h x h x dx h x dx
By the definition of the operator , passing to the limit in (86), it is easy to see that for any function from the definition domain of this operator, the following equality holds: (87) Finally, we obtain Now integrate Eq. (81) with account of formulas (82) and (88). We obtain the following asymptotic representations for its solutions when : where is an arbitrary real constant. By Theorem 3, with allowance for (67), for weak solutions of original problem (64)-(66), we have the following asymptotic representation when : Here, the function is determined by formulas (89), the function has the property when , and the function allows the estimation (where is a certain real number). Analyzing formulas (89) and (90), we come to the conclusion that for all solutions of problem (64)-(66) are bounded as (in the norm of the space ), while in the case , when , the solutions, generally speaking, are unbounded. Especially note that the existence of unbounded solutions is a consequence of the spatial heterogeneity of the perturbation factor in Eq. (64). In fact, suppose that in this equation . Then integral Eq. (26) with allowance for formulas (67) and (72), has the solution . Therefore, equation on the critical manifold (69) takes the form It is easy to show that all solutions of this equation have the following asymptotic representation as : where is an arbitrary real constant.

FUNDING
The reported study was funded by RFBR according to the research project 18-29-10055.