Journal of Function Spaces
Volume 2016 (2016), Article ID 5687920, 15 pages
http://dx.doi.org/10.1155/2016/5687920

Research Article

Nonclassical Problem for Ultraparabolic Equation in Abstract Spaces

Gia Avalishvili¹ and Mariam Avalishvili²

¹Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, 3 I. Tchavtchavadze Avenue, 0179 Tbilisi, Georgia
²School of Informatics, Engineering and Mathematics, University of Georgia, 77a M. Kostava Street, 0175 Tbilisi, Georgia

Received 6 March 2016; Accepted 24 April 2016

Academic Editor: Maria Alessandra Ragusa

Copyright © 2016 Gia Avalishvili and Mariam Avalishvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.

1. Introduction

Evolution equations with several time-like variables are encountered in various models of science and technology, particularly, in mathematical models of multiparameter Brownian motion [1], theory of boundary layers [2], mathematical models of diffusion of pollutants in water flows [3], transport theory [4], mathematical models of age structured biological population dynamics [5, 6], mathematical finance [7], mechanics, physics, and cosmology [8, 9]. In these models one of time variables is usual time and others might denote various quantities, for example, coordinates, temperature, and age or size of individuals of biological population.

Various mathematical problems are investigated for multitime evolution equations. Ultraparabolic equations with classical Cauchy conditions with respect to one time variable, problems with classical initial conditions with respect to all time variables, and classical or nonlocal boundary conditions with respect to space variables, inverse problems are investigated in spaces of classical smooth functions and in suitable spaces of generalized functions, and some papers are also devoted to numerical methods for ultraparabolic equations with classical initial conditions (see [10–26] and references given therein). Abstract rather general ultraparabolic equation was studied by Lions [27] in spaces of abstract vector-valued distributions. Applying methods of the theory of semigroups ultraparabolic integrodifferential equation with classical initial conditions with respect to time variables in the spaces of Hölder continuous vector-functions with values in a Banach space was investigated by Lorenzi [28] and ultraparabolic equations arising in age structured population dynamics with integral condition with respect to one of time variables were studied by many researchers; see [6, 29, 30] and their references.

Mathematical models of populations incorporating age structure [6] are given by ultraparabolic equations and include nonlocal integral condition with respect to age variable, that is, nonlocal initial condition with respect to the second time variable. Nonclassical problem with nonlocal initial condition for abstract first-order evolution equation was investigated by Krejn and Laptev [31] and in the case of parabolic equation was studied by Kerefov [32]. Various initial-boundary value problems with nonlocal initial conditions are generalizations of initial-boundary value problems with classical and periodical initial conditions and they can be also considered as problems of controllability by initial conditions. Later on, initial-boundary value problems with nonlocal initial conditions were investigated for various time-dependent partial differential equations and systems (see [33–45] and references given therein). Nonlocal in time problems with discrete nonlocal initial conditions are used for investigation of radionuclides propagation in Stokes fluid and problems of predicting the state of a medium [44, 45].

In this paper, we consider nonclassical problem for ultraparabolic equation:

with the following classical and nonlocal initial conditions:

where ,  ,  ,  , and are given vector-functions from suitable spaces, and is a given linear continuous operator in corresponding abstract Hilbert spaces.

As far as the authors know ultraparabolic equations with nonlocal initial condition (1) in abstract spaces have not been investigated yet. Moreover, in the case of classical initial conditions, that is, for , the well-posedness results are obtained either in spaces of continuous functions or in spaces of distributions with respect to time variables, for which the trace operator on boundary of domain of time variables cannot be defined. So, there are no results on the existence and uniqueness of solution in the spaces with minimal regularity properties necessary to define the initial conditions (2). In the present paper we obtain well-posedness theorem for nonlocal problem (1), (2), which in the case of also gives new existence, uniqueness, and continuous dependence result for classical problem for the abstract ultraparabolic equation.

The paper is organized as follows. In Section 2, we introduce some notations and give properties of spaces of square integrable vector-functions defined on two-dimensional domain with values in Hilbert spaces. We prove new trace theorem and formulas for integration by parts. In addition, we show characteristic properties of some spaces of vector-valued functions, which are used to investigate the nonclassical problem for ultraparabolic equation. In Section 3, we give variational formulation of the problem (1), (2) and prove our main result on the existence and uniqueness of its solution. Finally, in Section 4, we consider an application of the obtained abstract result to nonlocal problem for ultraparabolic partial differential equation in Sobolev spaces.

2. Preliminaries

We denote by the space of linear continuous operators from to , where and are Banach spaces. Let denote the space of continuous vector-functions on with values in . We denote by the space of measurable vector-functions such that and the generalized derivatives of are denoted by ,   ,  , for (cf. [46]), where is the space of infinitely differentiable functions with compact support in , where ,   is a bounded domain.

Throughout this paper, we use to denote generic constants that are independent of the main parameters involved, but whose values may differ from line to line and may change even within a single chain of estimates.

Let and be separable Hilbert spaces, such that is dense in and continuously embedded in it. We identify space with its dual by using inner product in and hence with continuous and dense embeddings, where is the dual space of [46]. We denote by the duality relation between and . To investigate problem (1), (2) let us consider the following space of vector-valued functions on :

which is a Hilbert space equipped with the norm

In the following three theorems we give some properties of the space .

Theorem 1. For each vector-function and ,  , there exist traces and , such that the trace operators and are continuous.

Proof. We can identify each vector-function from space with vector-function from . If , then , and since the linear combinations of products ,  , and are dense in , we have , . Note that with continuous and dense embeddings and . Consequently, from embedding theorem [46] we have that and is continuously embedded in

that is,

for any . Hence, we can define the trace operator , which is a linear continuous operator from to . Likewise, for each we obtain the existence of the linear continuous trace operator from to .

Theorem 2. For vector-functions the following formulas for integration by parts are valid:

Proof. For we consider the corresponding vector-functions defined in the proof of Theorem 1. For vector-functions the following formula for integration by parts is valid [46]:

where denotes the duality relation between and and denotes the inner product in . The latter formula is equivalent to the first formula of the theorem. Likewise we obtain the second formula.

Theorem 3. For vector-functions and the following equalities are valid:

Proof. Applying Theorem 2 for and ,  , , we obtain

for . Since ,  , we have , which proves the required equalities.

Let be a self-adjoint coercive operator, which means that the bilinear form satisfies the conditions

where . We also suppose that there exists a system of eigenvectors of the operator corresponding to the eigenvalues , that is, , for all , which is complete in and orthonormal in . From (11) it follows that the operator is invertible and [47]. Note that and hence , for all .

Let us define space , which is a Hilbert space equipped with the scalar product for all . Space is continuously embedded in , since

where . Space is dense in . Indeed, if , then . Since is dense in , there exists sequence ,  ,  , such that in , as . Consequently, in as and ,  .

So, with continuous and dense embeddings, where is the dual space of . The duality relation between and is denoted by . Note that for all we have

Hence, and . Therefore, the restriction of operator on is a linear continuous operator from to , and since is dense in , there exists unique continuous continuation of on , which we denote by ,   for all . From the definition of the operator it follows that

Indeed, since is dense in , there exists sequence ,  ,  , such that in as . Consequently, in as , and

In the following lemmas we give some properties of spaces ,  ,  ,  , and , which will be used to prove the main theorem.

Lemma 4. (a) If , then

(b) Suppose that ,  . The series converges if and only if the series converges in the space .

Proof. (a) Since the system of vector is orthonormal in we have

for all , and tending we obtain the inequality.
(b) The equivalence of the convergences of the series directly follows from the following equalities:

Lemma 5. (a) If , then

(b) Suppose that ,  . The series converges if and only if the series converges in the space .

Proof. (a) Since are eigenvectors of the operator and is orthonormal in we have

for all , and tending from (11) we obtain the required inequality.
(b) The equivalence of the convergences of the series directly follows from the following inequalities for all ,  ,

Lemma 6. (a) If , then

(b) Suppose that , . The series converges if and only if the series converges in the space .

Proof. (a) Let us define . Applying part (a) of Lemma 5 we have , where

Consequently,

(b) The equivalence of the convergences of the series directly follows from the following inequalities

Lemma 7. (a) If , then

(b) Suppose that ,  . The series converges if and only if the series converges in the space .

Proof. (a) Since are eigenvectors of operator and is orthonormal in we have for all , and hence

for all , and tending from the definition of the norm we obtain the required inequality.
(b) The equivalence of the convergences of the series directly follows from the following inequalities:

Lemma 8. (a) If , then

(b) Suppose that , . The series converges if and only if the series converges in the space .

Proof. (a) Note that the operator is linear and continuous, and applying (14) we obtain that satisfies the following coerciveness condition:

Consequently, the operator is invertible and [47].
Let us define . Applying part (a) of Lemma 7 we have and from equality (14) we obtain

for all . Consequently,

(b) Since , we have and the equivalence of the convergences of the series directly follows from the following inequalities:

for all ,  .

3. Main Result

In the present paper we investigate nonlocal problem (1), (2) applying the following variational formulation: find a vector-function ,  ,  , which satisfies equation

in the sense of distributions on , and initial conditions

in the spaces and , respectively. Applying Theorem 3 and by taking account of the density of linear combinations of products ,  ,  , in we obtain that (35) is equivalent to (1), which is considered as equation in the space .

Let us prove the following lemmas, which will be used to obtain the existence of solution of problem (34), (35). We denote by and the parts of the rectangle above and below the bisector , respectively; and are closures of and .

Lemma 9. If and , then the following estimates are valid

Proof. Taking into account the definition of the sets and we obtain

which implies the first estimate. Likewise we obtain the second estimate.

Lemma 10. If , then the following estimates are valid

Proof. Assuming that ,  , we have

From continuity of the trace operator we obtain

The latter inequality implies the first estimate of the lemma for functions from and since is dense in , by passing to the limit we obtain the estimate for functions from . Likewise we obtain the second estimate.

Lemma 11. If , then the following estimates are valid

where .

Proof. Assuming that we get

The continuity of the trace operator , , implies that

From the latter inequality we obtain the first estimate of the lemma for functions from and since is dense in , by passing to the limit we obtain the estimate for functions from . Likewise we obtain the second estimate.

The following existence and uniqueness theorem is valid for nonlocal problem (34), (35).

Theorem 12. If ,  ,  ,  , and satisfy compatibility condition and , then problem (34), (35) has a unique solution and the following estimate is valid:

Proof. Applying embedding theorem [46] we have that ,   and the compatibility condition is the equality in . First, we prove the existence of a solution. Let us consider the sequence of approximate solutions , where is a solution of the following nonlocal problem for the first-order partial differential equation:

where ,   ,  , . Note that ,  ,   and hence ,  .
Let us denote by and the following functions:

Functions and are defined for all and , respectively, because the trace of on each line

which is parallel to the bisector , which belongs to .
Note that and . Their generalized derivative is given by

for almost all ,

for almost all . Applying Lemma 10 we obtain

Therefore, and .
Applying Lemma 11 and continuity of the trace operator we obtain

Hence, and for , and from embedding theorem we obtain that ,   for .
Note that the classical problem for (45) with initial conditions and ,  ,  , has a unique solution ,  ,  , which is given by

From nonlocal conditions (46) for function we have

from which we obtain expression of for ,  ,

for ,  , where , if , and , if ;   denotes the integer part of .
Since functions ,  ,  , and are continuous, we have . Let us assume that and then . Note that ,   and from compatibility condition we have . Hence, we get

which imply continuity of at point . Moreover, we have

Consequently, and its generalized derivative belongs to , since

for almost all and

for and for almost all ,  .
Since and , from construction of it follows that satisfies conditions (46).
Note that the trace on the bisector ,  , of restriction of function on equals the trace on the bisector of restriction of function on , because ,  ,  , and

The generalized derivatives of function are given by

Therefore, we have and satisfies (45) in .
Now we obtain estimates for , which permit one to prove the convergence of the sequence of approximate solutions .
Applying Lemmas 9 and 10 we obtain

To obtain estimate for without loss of generality we assume that . Applying Lemmas 9–11 we get

From the latter estimate we have

Applying expressions for and , Lemma 9, and estimates (51)–(54) we obtain

In order to estimate without loss of generality we assume that . Applying Lemmas 9–11 we have

From the estimate for and (69) we get

By using Lemma 9 and estimates (52), (54), (66), and (69) we obtain

Applying parts (a) of Lemmas 4, 6, and 8 and inequalities (67), (70), and (71) we obtain

Consequently, from parts (b) of Lemmas 5 and 6 we get the sequence and hence the series

converges in the space . Since the trace operators ,   , and are continuous, from (46) we obtain that satisfies conditions (35), and, by (72), the estimate (44) is valid. Moreover,

and since the operator of differentiation is continuous on the space of distributions we have

Applying part (a) of Lemma 8 we obtain

and by using parts (a) of Lemmas 4–6 and (69) we get

Consequently, from part (b) of Lemma 8 we obtain

Let us show that ,  ,   is a solution of (34). Indeed, since the system is complete in , it follows that for each there exists a sequence , where is a linear combination of , such that in as . By taking account of the properties of eigenvectors , from (45) for each and we have

By passing to the limit in the last relation, we find that satisfies (34). So, the existence of solution is proved.
To complete the proof, we prove the uniqueness of solution. Note that the classical problem for (34) with initial conditions

has at most one solution. Indeed, if is a solution of the homogeneous classical problem (34), (80), that is, ,   and , then, by multiplying (1), which is equivalent to (34), by , integrating on , and applying Theorem 2 we get

and, consequently, in .
Since problem (34), (35) for is a classical problem, from the proof of the existence it follows that the classical problem for (34) with initial conditions (80), ,  ,  , ,  , has a unique solution ,  ,  ,  ,  , which can be represented in the form of convergent in series (73), where is given by (56). Therefore, the uniqueness of the solution of nonlocal problem (34), (35) follows from the uniqueness of the solution of problem (45), (46), which completes the proof.

4. Application

In this section, we consider an application of Theorem 12 obtained for abstract nonlocal problem to nonclassical problem for ultraparabolic partial differential equation with nonlocal initial condition. Let ,  , be a bounded domain with Lipschitz boundary [47]. By we denote the Sobolev space of functions in whose generalized partial derivatives up to the order belong to . We denote the closure in of the set by and its dual space by . We consider the second-order elliptic operator:

where ,  , and the following inequalities are valid

for almost all ,   () are arbitrary real numbers. Partial derivatives in the definition of are treated as generalized derivatives with respect to the corresponding variables. Note that the first-order partial derivatives of a function in belongs to the space . Since the multiplication of a function in the space by a function in leaves it in , for each function we have . The elliptic operator is a continuous operator from to the space , which, by virtue of (83) and Poincare inequality, satisfies the following self-adjointness and coerciveness conditions:

where and   is the duality relation between the spaces and . We denote by the domain of operator . The bilinear form corresponding to the operator is of the form

and satisfies conditions (11). Since the continuous embedding of in is compact, it follows from general theorem for self-adjoint coercive operators mapping a Hilbert space into its dual [47] that there exists a system of eigenfunctions of the operator corresponding to the eigenvalues , such that is orthonormal in and complete in , and ,   as .

Let us consider nonclassical problem for ultraparabolic partial differential equation

with nonlocal initial and homogeneous boundary conditions

where . We identify functions defined on with vector-functions defined on and ranging in the corresponding function spaces defined on . Hence, the nonlocal in time problem (86)–(88) can be stated as the problem of finding a vector-function ,  ,  ,  , which satisfies (86) in the space of distributions on ranging in , where the partial derivatives and are treated as the first-order generalized derivatives . The unknown function satisfies conditions (87) in the spaces and , respectively. The boundary conditions (88) are valid, because the trace of function from vanishes on the boundary of the domain . Thus, nonlocal in time problem (86)–(88) is a particular case of the nonclassical problem (34), (35) for ,   , and . The following existence and uniqueness theorem is valid for nonlocal in time problem (86)–(88).

Theorem 13. If ,  ,  , , and satisfy compatibility condition for all and , then nonlocal problem (86)–(88) has a unique solution , , , , and the following estimate is valid:

5. Conclusions

In this paper, we investigated the existence and uniqueness of solution of nonclassical problem for ultraparabolic equation with two time variables and nonlocal initial condition, which connect values of the unknown vector-function at the initial and some later moment of time. The considered problem is a generalization of the classical problem for ultraparabolic equation and the obtained results give new well-posedness result for the classical problem, as well as for nonlocal one, in suitable function spaces with minimal regularity that is necessary to define the traces on the boundary of the time domain. Note that applying the results obtained in this paper and the presented method of investigation one can study various initial-boundary value problems for ultraparabolic equations with several time variables and more general nonlocal initial conditions. The proof of the existence of solution of the nonclassical problem is constructive and can be used to obtain approximate solutions of the nonlocal problem.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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