Nonlinear Equations of Infinite Order Defined by an Elliptic Symbol

The aim of this work is to show existence and regularity properties of equations of the form f(Δ)u = U(x, u(x)) onR, in which f is a measurable function that satisfies some conditions of ellipticity and Δ stands for the Laplace operator on R. Here, we define the class of functions to which f belongs and the Hilbert space in which we will find the solution to this equation. We also give the formal definition of f(Δ) explaining how to understand this operator.


Introduction
This paper is motived by recent researches in string theory and cosmology where the equations appear with infinitely many derivatives [1][2][3][4][5][6][7][8][9][10][11][12][13].For example, we can mention the following equation: where  is a prime number.This equation describes the dynamics of the open -adic string for the scalar tachyon field (see [4,7,8,[10][11][12] and the references therein).To consider this equation as an equation in an infinite number of derivatives, we can formally expand the left-hand side as a power series in  2  .Let us note that, in the articles [10,11], (1) has already been studied via integral equation of convolution type and it is worth mentioning that in the limit  → 1 this equation becomes the local logarithmic Klein-Gordon equation [14][15][16].
Another common example of an equation with infinitely many derivatives that is worth pointing out corresponds to the dynamical equation of the tachyon field in bosonic open string field theory that can be set as where ◻ = − 2  + Δ is the d' Alembertian operator (see [17]).
In the present paper, our aim is to show existence and regularity of solutions for nonlinear equations of infinite order of type  (Δ)  =  (⋅, ) , where the operator (Δ) is defined in terms of Laplacian over R  and the function  is defined in whole Euclidean space R  .First, we define the class of functions to which the symbol  belongs.This class, as we will see, contains symbols that are from a very general kind and in general do not belong to the classic Hörmander class defined to pseudodifferential operators [18].It is worth pointing out that this paper is inspired by the articles [19][20][21], where the authors work out this type of equations.In the article [19], the authors consider the operator (Δ) acting on whole Euclidean space R  or over a compact Riemannian manifold (, ) and show the existence and regularity of solutions for certain values of a constant  > /2, where  > 1, which will be defined in detail in Section 2. In the present paper, assuming that the nonlinearity  satisfies a Lipschitz type inequality, we extend these results and show the existence and uniqueness of solutions for (3) for values of  > 0.
This work is organized as follows: in Section 2, definitions and basic properties about the class of functions to which the symbol  belongs are given.Furthermore, we introduce the vector space where we will seek the solution to nonlinear 2 International Journal of Mathematics and Mathematical Sciences equation (3).At the end of this section, the definition of operator (Δ) and an embedding lemma are given.In Section 3, we solve the linear equation where  ∈  2 (R  ), obtaining also some properties that the solutions of this equation have and which will be useful to solve the nonlinear equation in the following section.Finally in Section 4, using Banach's fixed point theorem, we show existence and uniqueness of the solution to the nonlinear problem (3).

Preliminaries
The aim of this section is to define and develop some basic aspects that will be needed in the study of the nonlinear equation If a measurable function satisfies the above two conditions, we will say that  belongs to the class G  or simply that  is a G  -symbol.Although the above condition (  ) coincides with the condition of ellipticity given for pseudodifferential operators (see [22]), we can highlight that these symbols are not defined symbols in the sense of Hörmander [18].Now, from the definition of class G  , we obtain the following propositions.
Multiplying both inequalities, we obtain that where . On the other hand, let  ∈ G  ; then  satisfies condition (P) and there exist real numbers , , and  such that for all  with || >  we have that thus  ∈ G  .

Lemma 3.
Let  > 0 be fixed, and let Then   ∈ G  for all  ≤ .
Proof.Clearly the function Now, in order to see that   satisfies the condition (  ), let us note that, for  < , we have Let us note that, in the right-hand side of the above equality, Therefore, for   < 1 fixed, there exists   > 0 such that, for all  with || >   , we have Finally, we have   ∈ G  for  ≤ .
Next, we introduce the vector space where we will find the solution to our nonlinear equation.Definition 4. Given  > 0 and the symbol  in the class G  fixed, one defines the space H  () as the set of complex valued functions  defined on R  such that  is measurable, its Fourier transform F() exists, and We can endow H  () with the following inner product.
and with this definition, the vector space H  () turns out to be a Hilbert space.Moreover, from the definition of H  () International Journal of Mathematics and Mathematical Sciences 3 and Plancherel's theorem, we have that H  () →  2 (R  ) since Proposition 5. Let   be defined as in Lemma 3. Then a function  ∈ H  (  ) if and only if  ∈   (R  ).
Proof.First, let us note that In the last equality, we have used the notation given by Taylor [23].Next, by the definition of H  (), we have that  ∈ H  (  ) if and only if there exist its Fourier transform F() and This is equivalent, as we have seen above, to and this is equivalent to  ∈   (R  ).
We now introduce the definition of operator (Δ).The reason why we will consider this definition comes from a formal computation; see, for instance, [19].Definition 6.For a G  -symbol , one defines the operator (Δ) as follows: Analogously, we can define the linear operator  = (Δ)+  as It is easy to see that  acts on H  () and for all  ∈ H  () we have that  ∈  2 (R  ).

The Linear Equation
In this section, we will consider the linear operator  defined in (19).We solve the linear equation where  ∈  2 (R  ).Furthermore, we establish certain regularity properties that enjoy the solutions of the linear equation (20).
Theorem 8. Let  be in the class G  .Then, for each  ∈  2 (R  ), there exists a unique solution   ∈ H  () to linear equation (20).Moreover, the equality holds.
Proof.By the definition of operator  given in (19), we have that the equation  =  is equivalent to Now, since  ∈  2 (R  ), we can apply Fourier transform to both sides of the above identity, obtaining that Applying the inverse Fourier transform to both sides of this equality, we find the explicit form of Hence is the unique solution of the linear equation  = .In addition, we have that International Journal of Mathematics and Mathematical Sciences Now, in the following two propositions we will show that some extra properties of  will imply additional regularity of .
Proof.As we have seen, if  is the solution to the linear equation (20) thus,  ∈ H  (ℎ).
Now, we will show that if the function  in ( 20) is invariant under rotations, then the solution  will be invariant under rotations too.Proposition 10.If  is invariant under rotations, that is, for each rotation,  ∈ () and for all  ∈ R  , () = () holds, then the solution  to the linear equation  =  is invariant under rotation as well.
Proof.Suppose that () = () where  ∈ (); then let us note that the Fourier transform of  is invariant under rotation too.Indeed,

The Nonlinear Equation
In this section, our aim will be to study the nonlinear equation where the nonlinearity is given by the function where  is a nonnegative constant.Assuming certain growth condition for the function , we will prove the existence and uniqueness of solution to this equation.For this purpose, our main tool will be Banach's fixed point theorem and the results developed in Section 3.
Theorem 11.Let  ∈ G  .For  > 0, consider the function   defined by where  is a function such that (⋅, 0) ∈ If we consider the function  given by (35), then the nonlinear equation ( 34) is equivalent to in which  is defined by (19).Now, let us define the operator R : where ũ is the unique solution to the linear equation ũ = (⋅, ).
From this, we see that Then, and since we get Now, if we choose sufficiently small , such that ‖ℎ‖  ∞ (R  ) < 1, then we have that R is a contraction, and by Banach's fixed point theorem, there exists a unique  0 ∈ H  () such that That is, It is easily seen that this function satisfies the conditions of Theorem 11 and for all  ∈ H  (), we have (⋅, ) ∈  1 (R 2 ), and then applying Corollary 12 with  = 3,  = 1, and  = 2, we have that  = 2; therefore, the solution to nonlinear equation (34) with the nonlinearity (50) belongs to the class  2 (R 2 ).

Proposition 1 .
Let ,  > 0 be fixed.If  ∈ G  and  ∈ G  , then  ⋅  ∈ G + .Proof.It is clear that the function  ⋅  satisfies condition (P).On the other hand, since  and  satisfy condition (  ), we have that there exist positive constants  1 ,  2 , , ,  1 , and  2 such that 2(R  ).Now, let us see that if  ∈ H  (), then the function (⋅, ) defined over R  is a function belonging to  2 (R  ).Indeed,