Parabolic Equations of Infinite Order with Data

Abstract

We prove an existence result of a nonlinear parabolic equation under Dirichlet null boundary conditions in Sobolev spaces of infinite order, where the second member belongs to .

1. Introduction

This paper is devoted to the study of the following strongly nonlinear parabolic problem of Dirichlet type in the cylinder : where is a bounded open subset of and is a cylinder with lateral surface , with is the boundary of . is a nonlinear elliptic operator of infinite order defined by Such operators include as a special case Leray-Lions types in the usual sense.

The real functions are assumed to satisfy some growth and coerciveness conditions without supposing a monotonicity condition in , for all multi-indices .

The nonlinear term satisfies natural growth on and has to fulfil a sign condition.

The data is assumed to satisfy In the case of infinite order, Dubinskiĭ [1] has proved, under some growth hypothesis and certain monotonicity conditions, the existence of solutions for the Dirichlet problem associated with the equation in some general functional Sobolev spaces of infinite order of variables exponents , with being a multi-indice. The same author has investigated the existence result for parabolic elliptic problems governed by operators of infinite orders. In fact, also in [1], Dubinskiĭ has proved by considering, further, the monotonicity of the operator that the problem has a solution in ,  , in the variational case (i.e., where belongs to the dual space).

Another work has been shown, in the variational case in [2], the existence of solutions for strongly parabolic nonlinear equations of infinite order related to the problem .

Our purpose in this paper is to prove the existence of solutions for parabolic equations, in Sobolev spaces of infinite order with data, associated with the problem .

More precisely, we will assume more less restrictions on the operator (no monotonicity condition) and deal with a different approach by involving a truncation of the perturbations . Next, we use the monotonicity of a part of approximate operator which contains a linear term of higher order of derivation that satisfies the monotonicity condition and prove the existence of solutions in the framework of function space ,  .

Let us mention that an interesting result concerning the stationary counterpart of the problem has been proved in [3, 4].

2. Preliminaries

Let be a bounded domain in ,  ,   real numbers for all multi-index , and the usual Lebesgue norm in the space . The Sobolev space of infinite order is the functional space defined by Here .

We denote by the space of all functions with compact support in with continuous derivatives of arbitrary order.

Since we will deal with the Dirichlet problem, we will use the functional space defined by In contrast with the finite order Sobolev space, the very first question, which arises in the study of the spaces , is the question of their nontriviality (or nonemptiness), that is, the question of the existence of a function such that .

Definition 1 (see [1]). The space is called nontrivial space if it contains at least one function which is not identically equal to zero; that is, there is a function such that .

It turns out that the answer of this question depends not only on the given parameters and of the spaces , but also on the domain .

The dual space of is defined as follows: where for all multi-indices and is the conjugate of ;  that is,   (for more details about these spaces, see [1, 5]).

By the definition, the duality of the space and is given by relation which, as it is not difficult to verify, is correct.

Let us denote by the space of functions which has finite norm and is equal to zero together with all derivatives on the lateral surface . In other words one has Further, let be the dual space of the space , that is, the space of generalized functions having a form where and The value of on an element is defined by the formula which, as easy to see, is correct.

Sobolev spaces of infinite order have extensive applications to the theory of partial differential equations and, among their number, in mathematical physics. The basis of these applications is the nonformal algebra of differential operators of infinite orders as the operators, acting in the corresponding Sobolev spaces of infinite order. This makes it possible, by considering as a parameter, to solve a partial equation as ordinary differential equation, to which are adjoined the initial or boundary conditions.

More explicitly, we cite the following examples of operators of infinite order which are closely inspired from the ones used in Dubinskiĭ [1].

Example 2. Consider the following operator: Our technique here consists in exploiting certain results in the setting of functional spaces of infinite order. Thus as in [1], we can write the operator as follows: where , , are real numbers which guarantee the nontriviality of the corresponding functional space defined by Moreover for any , the parabolic problem has a solution , in the variational sense.
By using the recent work of authors (see Theorem 3.1. in [2]), the strongly nonlinear parabolic problem has also a solution , in the variational sense, for any . Here is a nonlinear term which has to fulfil a sign condition (see Section 3).

Remark 3. For examples of the nontriviality of Sobolev spaces of infinite order, we refer the reader to [1, 5–7] for details.

3. Main Result

In this section we formulate and prove the main result. We denote by the number of multi-indices such that . Let be the nonlinear operator of infinite order defined as in (1), with being a real function.

Let us now formulate the following assumptions. is a Carathéodory function for all .For a.e. , all , all , and some constant , we assume that  where ,   are real numbers for all multi-indices .There exist constants ,   such that  for all , for all and for a.e. .The space is nontrivial. As regards the nonlinear term , we assume that satisfies the following natural growth on and the classical sign condition.  is a Carathéodory function satisfying  for a.e. ,  , and some constants and . Concerning the second member , we assume that  We will prove the following existence theorem.

Theorem 4. Under assumptions and , for any right side there exists at least a function such that(1), ;(2);(3)for any function , the following identity
 is valid.

Proof. we proceed by steps in order to prove our result.
Step  1. The approximate problem.
Set for a.e. where is the usual truncation given by It is clear that for a.e. . Thus, it follows that .
Further, we have and from Lebesgue’s dominated convergence theorem, see [8], we conclude that Let sufficiently large. Define the operator of order by Note that are constants small enough such that they fulfil the conditions of the following lemma introduced in [1].
In fact, such a condition imposed on each is required to ensure the nontriviality of the space .
Lemma  5 (cf. [1]). For any nontrivial space there exists a nontrivial space such that .
The operator is clearly monotone since the term of higher order of derivation is linear and satisfies the monotonicity condition (see [1, 3]). Moreover, thanks to the truncation as in [9] and from assumptions , , and , we deduce that the operator is bounded, coercive, and pseudo-monotone. Then, it is well known (see Lions [10]) that there exists such that In the variational formulation, we get for any .
Step  2 (a priori estimates). Let us choose as a test function in . Then using the sign condition in , one has the estimates In the sequel   designate arbitrary constants not depending on .
From the first equality in and estimates (28) and (29), we remark that . In addition, for any the following equality is valid: where
Regarding the quantity , one has and so We also have where is the constant of the estimate (28). Then one gets Moreover, for the last term , one has where Then, one deduces that Combining (30), (33), (35), and (38), it follows that This implies that that is, the derivatives form a bounded set in the space .
Now, estimates (28) and (40) permit us to apply the well known lemma of compactness (see Lions [11]).
Let , , and be Banach spaces. Let us set where , are real numbers.
Lemma  6 (cf. [1]). Let the imbeddings hold; moreover, let the imbedding be compact. Then and this imbedding is compact.
In order to apply this lemma, define where is arbitrary and .
Step  3 (convergence of the approximate problem ). In view of (28) and (40), we deduce that the family of solutions of problems is compact in the space , where is arbitrary. Consequently, by similar argument as in the elliptic case (using the diagonal process), see [3] or [1], one gets that the sequence converges strongly together with all derivatives to a function .
Letting now be fixed, a measurable subset of , and , we have where is the constant of (29) which is independent of .
For sufficiently small and , we obtain Using Vitali’s theorem, we get On the other hand, in view of Fatou’s lemma and (29), we obtain this implies that Now, we will prove that for all .
In fact, let be a fixed number sufficiently large and let . Set where or, in another form, with and for ( are constants given in Lemma 5).
We will go to limit as to prove that , , and tend to . Starting by , we have since is of Carathéodory type.
The term is the remainder of a convergence series; hence .
For what concerns , for all , there holds (see [8, page 56]) such that where is the constant given in the estimate (28). Moreover the term is the remainder of a convergent series; therefore holds.
Finally, we conclude that for all .
Moreover, it is clear that as since .
Consequently, by passing to the limit in , we obtain for all .
That is, for all .
This completes the proof.

4. Example

The following example of an operator of infinite order is closely related to the one used in [12].

Let us consider the operator: where is a sequence of numbers, is a number such that the space is not trivial (e.g., if and ); then the conditions ,  , and are satisfied.

As regards a function that satisfies the condition , let us consider where , , a.e.

Consequently, for the described above operator and the nonlinear term   the existence result follows immediately from Theorem 4.

Conflict of Interests

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

Acknowledgments

S. El Manouni is grateful to the Department of Mathematics, Technische Universität Berlin, Germany, and in particular to Professor Karl-Heinz Förster for his hospitality. The research of S. El Manouni is supported by the Arab Fund for Economic and Social Development (AFESD), Kuwait.