Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables

Abstract We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.


Introduction
The problem of finding the solution T .t / of ordinary differential equation (ODE) of order n 2 Nnf1g, which satisfies conditions T .t 1 / D c 1 ; : : : ; T .t n / D c n , where t 1 < : : : < t n ; t 1 ; : : : ; t n ; c 1; : : : ; c n 2 R, in the literature is found as the Vallee-Poussin problem or multipoint (n-point) problem.
For the first time such problems were studied in the articles [1][2][3][4], in which the importance to study the problems is indicated from the point of view of generalization of Cauchy problem. In contrast to Cauchy problem, the multipoint problem is ill-posed, because corresponding homogeneous problem can have nontrivial solutions.
The first results on solving the problems with multipoint time conditions for linear partial differential equations (PDE) have been obtained in [5] based on the metric approach. In particular, this paper points out the problem of small denominators, which is typical for multipoint problems. Also, ill-posedness of these problems was proved, morover it was shown that the classes of uniqueness of the multipoint in time problem solutions for PDE were significantly different from the classes of uniqueness of the solutions of the corresponding Cauchy problem for the same equations.
The investigation of the n-point problem for equations and systems of PDE's in the bounded domains that is based on the metric approach has significantly developed in recent years (see works [6,7] and bibliography).
Papers [8][9][10] are devoted to establish the classes of unique solvability of problems with multipoint conditions in time for PDE's in unbounded domains (strip, layer). The technique of investigation multipoint problems in spaces of functions in which there is no problem of small denominators is proposed in these works.
The solvability of problems with multipoint conditions for differential-operator equations was studied in [11].
This article is devoted to study of the null space of problem for the PDE of the second order with respect to time variable, and arbitrary order with respect to spatial variables with local two-point conditions in time, and it is a continuation of researches [12] for the case of equations with several spatial variables.

Problem statement
In the domain R 1Cs ; s 2 Nn f1g ; we investigate the set of solutions U D U.t; x/, x D .x 1 ; : : : ; x s /, of the PDE in which the operator coefficients a @ @x Á , b @ @x Á , @ @x D @ @x 1 ; : : : ; @ @x s ; are considered as arbitrary differential expressions with complex coefficients of the finite or infinite order, and the symbols a . /, b . / of those coefficients are entire functions of complex vector-variable 2 C s .
Under condition that a . / and b . / are polynomials, denote their degrees by the set of variables as p a 2 Z C and p b 2 Z C . Also we assume that p a D 1 and p b D 1 if a . / and b . / respectively are not polynomials.
In the solutions set of equation (1), we will find the solutions that satisfy the homogeneous local two-point conditions: @ @x Á are differential polynomials with complex coefficients. Moreover their symbols A 1 . /, A 2 . /, B 1 . /, B 2 . / for 2 C s satisfy the conditions: In this article, we establish the conditions of existence of nontrivial solutions (nontrivial null space) of this problem.
In the case of existence of such solutions, we propose the way of their construction by using the differential-symbol method [13].

Main results
In equation (1), we replace @ @x with vector-parameter and the symbol @ @t with d dt . We obtain the ODE in which, from now on, we consider 2 C s . The normal fundamental system of solutions of equation ( We write down the determinant of the form: Such determinant will be called the characteristic determinant of problem (1), (2). It can also be represented in a matrix form: Note that function . / as a superposition of entire functions is an entire function. (1), (2) in the class of entire functions has only trivial solution.

The case when the set of zeroes of the characteristic determinant is empty
Proof. Let there exists a nontrivial integer solution U.t; x/ of equation (1), i.e. entire function of the form of variables t and x D .x 1 ; : Then '.x/ and .x/ are also entire functions. Write down the solution of problem (1), (2) according to differential-symbol method [13] as the solution of Cauchy problem for equation (1) with initial data ' and in the form: where x D 1 x 1 C : : : C s x s , O D .0; : : : ; 0/. Since T 0 .t; / e x and T 1 .t; / e x are entire in functions of order p D max fp; 1g 1, so both entire functions '.x/ and .x/ must have the adjoint [15, p. 316] with p order q, i.e. q D p= .p 1/ for 1 < p < 1, and q D 1 for p D 1 and q D 1 for p D 1.
Since two-point conditions (2) for function ' and are satisfied, we obtain in R s the system of identities where onto the first identity of system (5) and act by A 2 @ @x Á onto the second identity of the system, after that subtract the second identity from the first one. We obtain Since the function 1 . / is entire, so, acting by expression 1 @ @x Á onto the last identity, we obtain Similarly, if we act by expression B 1 @ @x onto the first identity of system (5) and by expression A 1 @ @x Á onto the second identity, and then add the obtained identities, we get We have the identity U .t; x/ Á 0, which contradicts to the assumption on nontriviality of the solution of problem (1), (2).

The case when the set of zeroes of the characteristic determinant coincides
Consider the functionˆ.
which is an entire function with respect to the vector-parameter 2 C s . Let's denote the order of this function in the set of parameters 1 ; : : : ; s by e p. Note that 1 Ä e p Ä p. (2), then entire nontrivial solutions of the problem exist, and they can be represented in form where '.x/ is an arbitrary entire function, that has order, adjoint with the order e p.
Proof. Since . / Á 0 then there holds the equality Let's show firstly that function (7) satisfies equation (1). Taking into account the commutativity of the operations @ @t , @ @x and @ @ , we obtain Let's prove that the first condition (2) is satisfied: Due to equality (8), function (7) also satisfies the second condition (2): Thus, we have proved that function (7) is a solution of problem (1), (2), in which '.x/ is an arbitrary entire function with the order adjoint to the order e p.
Let's show that function (7) is nontrivial. In fact, Since jA 1 . /j 2 C jA 2 . /j 2 > 0 for arbitrary 2 C s , thus either U.0; x/ or @U @t .0; x/ are nontrivial. Hence, function (7) is also nontrivial as a solution of Cauchy problem with nonzero initial data. Besides the set (10) let us consider such sets:

The case when the set of zeroes of the characteristic determinant is not empty and does not coinside with C s
Note the following fact: if ! 2 .˛/, then . / .˛/ D 0 for all 2 Z s C for which Ä !.
Proof. The function U .t; x/ is the solution of equation (1). It follows from equality (7) for entire function '.x/ D x ! . The first condition from two-point conditions (2) is fulfiled:

0:
Let's show that the second condition in (2) is satisfied: Since ! 2 .˛/, then according to the remark for arbitrary q 2 Z s C such that q Ä !, equalities .q/ .˛/ D 0 are fulfiled. Hence, the function (11) satisfies the second condition (2).
Let's prove that the formula (11) defines nontrivial solution of problem (1), (2). Let's calculate the value of the function (11) and its derivative at the point t D 0: In the last sums, the coefficients of x ! are A 2 .˛/ and A 1 .˛/. From the condition jA 1 .˛/j 2 C jA 2 .˛/j 2 ¤ 0, it follows that at least one of the expressions U .0; x/ or @U @t .0; x/ is a nonzero quasipolynomial. This implies that function (11) is a nontrivial solution of problem (1), (2).

Examples
that satisfy local two-point conditions For this problem, we obtain a. / D 1 The normal at the point t D 0 fundamental system of solutions of the corresponding to (12) ODE Let's write down the characteristic determinant of problem (12), (13): In the case h ¤ k; k 2 N, the condition . / ¤ 0 is satisfied for arbitrary 2 C 2 , so by Theorem 3.1, problem (12), (13) has only trivial solution.
If h D k, where k 2 N, then there holds the equality . / Á 0. So, by Theorem 3.2, there exist nontrivial solutions of problem (12), (13). Function (6) gets the form: We search the nontrivial solutions of problem (12), (13) in the case h D k, where k 2 N, by formula (7): The null space of the problem is infinite-dimensional. Note that the condition of entirety of the function ' can be weakened, considering classical solutions of problem (12), (13): ' can be arbitrary twice continuously differentiable function on R 2 . 4 Example 4.2. In the domain .t; x 1 ; x 2 / 2 R 3 , find the solutions of the two-point problem For this problem, we have a. / The characteristic determinant of problem (14) and the corresponding set M will have the form: The set M consists of the following vectors: By Theorem 3.3, for arbitrary 2 C n˚2 mi; m 2 Z « and k 2 Z, we find such nontrivial solutions of problem (14): Note that the obtained solutions of problem (14) are linearly independent. 4 Example 4.3. In the domain .t; x/ 2 R 4 find the solutions of two-point problem in which a @ @x Á ; c @ @x Á are differential polynomials with complex coefficients.
Problem (15) where ' is an arbitrary entire function of three variables. The order of this function is adjoint with number p D max f1; deg a. /g (deg a. / defines degree of polynomial a. / by the set of variables 1 ; 2 , 3 ).
The set M consists of the following vectors:˛Á˛. 1 ; 2 / D . 1 ; 2 ; 0/, where 1 ; 2 2 C. We calculate The set .˛/ contains all multi-indexes ! 2 Z 3 C such that ! .0; 0; 1/. Since for arbitrary m; n 2 Z C equalities hold .m;n;0/ .˛/ D 0, then .˛/ D n .m; n; 0/; m; n 2 Z C o : Using the Theorem 3.3 for arbitrary m; n 2 Z C and 1 ; 2 2 C we find such nontrivial solutions of problem (16): where ' is an arbitrary nontrivial entire function of two variables, whose order in the set of variables is not greater than second, and 1 ; 2 are arbitrary complex parameters. 4

Conclusions
In this work we proved that the homogeneous problem for PDE of the second order in time variable, in which local two-point conditions are imposed, and generally infinite order in spatial variables, has only trivial solution in the case when the characteristic determinant of the problem is not equal to zero. In the other case, when the characteristic determinant possesses a nonzero value, we proved the existence of nontrivial solutions of the problem and proposed the differential-symbol method for their construction. We also gave some examples of using this method. The investigation of the null space, of the problem given herein, provide an opportunity for future to specify the classes of unique solvability of the corresponding nonhomogeneous problem.