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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

This paper is devoted to the study of the following strongly nonlinear parabolic problem of Dirichlet type in the cylinder

The real functions

The nonlinear term

The data

Another work has been shown, in the variational case in [

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

More precisely, we will assume more less restrictions on the operator

Let us mention that an interesting result concerning the stationary counterpart of the problem

Let

We denote by

Since we will deal with the Dirichlet problem, we will use the functional space

The space

It turns out that the answer of this question depends not only on the given parameters

The dual space of

By the definition, the duality of the space

Let us denote by

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

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

Consider the following operator:

By using the recent work of authors (see Theorem 3.1. in [

For examples of the nontriviality of Sobolev spaces of infinite order, we refer the reader to [

In this section we formulate and prove the main result. We denote by

Let us now formulate the following assumptions.

For a.e.

where

There exist constants

for all

The space

As regards the nonlinear term

for a.e.

Concerning the second member

We will prove the following existence theorem.

Under assumptions

for any function

is valid.

we proceed by steps in order to prove our result.

Set for a.e.

Further, we have

In fact, such a condition imposed on each

The operator

From the first equality in

Regarding the quantity

Now, estimates (

Let

In order to apply this lemma, define

Letting now

For

In fact, let

We will go to limit as

The term

For what concerns

Finally, we conclude that

Moreover, it is clear that

Consequently, by passing to the limit in

That is,

This completes the proof.

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

Let us consider the operator:

As regards a function

Consequently, for the described above operator

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

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.