Abstract
The aim of this work is to show existence and regularity properties of equations of the form on , in which is a measurable function that satisfies some conditions of ellipticity and stands for the Laplace operator on . Here, we define the class of functions to which belongs and the Hilbert space in which we will find the solution to this equation. We also give the formal definition of explaining how to understand this operator.
1. Introduction
This paper is motived by recent researches in string theory and cosmology where the equations appear with infinitely many derivatives [1–13]. 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, 10–12] 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 . 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 this equation becomes the local logarithmic Klein-Gordon equation [14–16].
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 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 where the operator is defined in terms of Laplacian over and the function is defined in whole Euclidean space . 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 [19–21], where the authors work out this type of equations. In the article [19], the authors consider the operator acting on whole Euclidean space or over a compact Riemannian manifold and show the existence and regularity of solutions for certain values of a constant , where , 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 .
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 equation (3). At the end of this section, the definition of operator and an embedding lemma are given. In Section 3, we solve the linear equation where , 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).
2. 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 defined in terms of the “symbol .” First, we give sense to operator . For this purpose, we have to clearly define the conditions that should be satisfied by the symbol . Thus, we give the following definition.
Let us consider measurable functions such that the following two conditions are verified: (P) the function is nonnegative; there exist real numbers , , and such that If a measurable function satisfies the above two conditions, we will say that belongs to the class or simply that is a -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 , we obtain the following propositions.
Proposition 1. Let , be fixed. If and , then .
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 , , , , , and such that Multiplying both inequalities, we obtain that where and . Therefore .
Proposition 2. If , then .
Proof. Since and , then . On the other hand, let ; then satisfies condition (P) and there exist real numbers , , and such that for all with we have that thus .
Lemma 3. Let be fixed, and let . Then 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 fixed, there exists such that, for all with , we have
Finally, we have for .
Next, we introduce the vector space where we will find the solution to our nonlinear equation.
Definition 4. Given and the symbol in the class fixed, one defines the space as the set of complex valued functions defined on such that is measurable, its Fourier transform exists, and
We can endow with the following inner product. Given , , and with this definition, the vector space turns out to be a Hilbert space. Moreover, from the definition of and Plancherel’s theorem, we have that since
Proposition 5. Let be defined as in Lemma 3. Then a function if and only if .
Proof. First, let us note that
In the last equality, we have used the notation given by Taylor [23]. Next, by the definition of , we have that if and only if there exist its Fourier transform and .
This is equivalent, as we have seen above, to
and this is equivalent to .
We now introduce the definition of operator . The reason why we will consider this definition comes from a formal computation; see, for instance, [19].
Definition 6. For a -symbol , one defines the operator as follows:
Analogously, we can define the linear operator as It is easy to see that acts on and for all we have that .
Lemma 7. Let ; then (1)for and , the embedding holds;(2)for all such that , the embedding holds.
3. The Linear Equation
In this section, we will consider the linear operator defined in (19). We solve the linear equation where . Furthermore, we establish certain regularity properties that enjoy the solutions of the linear equation (20).
Theorem 8. Let be in the class . Then, for each , there exists a unique solution 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 , 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 then But then, from the definition of and Plancherel’s theorem, we have that and
Now, in the following two propositions we will show that some extra properties of will imply additional regularity of .
Proposition 9. Let , consider the linear equation , defined in (19), and let . If, in addition, for some for some , then the solution .
Proof. As we have seen, if is the solution to the linear equation (20), then
holds. Multiplying both sides of the above equation by and integrating over , we have obtained
Since , then we have that
thus, .
On the other hand, since , we have
Therefore, .
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 , 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, Hence,
4. 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 . For , consider the function defined by where is a function such that . If there exists a function , such that the following inequality holds: then for sufficiently small , there exists a unique solution to problem (34).
Proof. Note that, from condition (37), we have for the function the following estimate:
Now, let us see that if , then the function defined over is a function belonging to . Indeed,
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 by
where is the unique solution to the linear equation .
From this, we see that
Now, from the linearity of ,
and due to Theorem 8, we have that
Then,
and since
we get
Now, if we choose sufficiently small , such that , then we have that is a contraction, and by Banach’s fixed point theorem, there exists a unique such that
That is,
Corollary 12. Let , , , and be as in Theorem 11. In addition, suppose that the function is such that there exist real constants , with , so that for all one has . Then, the solution to nonlinear equation (34) with the nonlinearity (35) belongs to the class .
Proof. Let be the operator defined in (41). Since is acting over and for all we have , then, by Proposition 5, . Subsequently by Proposition 9, we have that, for all , , as , and by Lemma 7, we get . Therefore, the fixed point of , that is, the solution to nonlinear equation (34), belongs to the class .
Let us see the following example.
Example 13. Let and defined by It is easily seen that this function satisfies the conditions of Theorem 11 and for all , we have , and then applying Corollary 12 with , , and , we have that ; therefore, the solution to nonlinear equation (34) with the nonlinearity (50) belongs to the class .
In Theorem 11, if the function satisfies a Lipschitz-type condition with respect to the variable , then there exists a unique solution to the nonlinear equation (34). Now, we will see that if we change the global Lipschitz condition to a local Lipschitz condition only, for certain , the nonlinear equation has a solution, but we cannot ensure uniqueness.
Theorem 14. Let and . Consider the function defined by where . Suppose that, for all , there exists such that, for all , the inequality holds. Then, for sufficiently small , there exists solution to problem (34).
Proof. Let ; then
and if , then we have
Now, let us define the following set:
and the operator by , where is the unique solution to the linear equation .
Let us see that, for all , if we consider
we have , because is the solution to the linear equation ; then we have
Now, let us note that
Next, if we choose , we have that is a contraction; therefore, choosing by Banach’s fixed point theorem, we have that there exists a unique solution which is a solution to the nonlinear equation (34).
Now, we will see that if the nonlinearity is radial, that is, invariant under rotations with respect to , then the unique solution to the equation is radial as well. To show this, we define the set Note that is a closed set in . Thus is a Hilbert space.
If we consider in the metric , we can clearly see that is a complete metric space.
Theorem 15. Suppose that , and that the nonlinearity is invariant under rotations with respect to . Assume also that and there exists such that Then, for sufficiently small , there exists a unique solution to the nonlinear equation (34).
Proof. Let be the operator defined by , where is the unique solution to the linear equation . Since the nonlinearity is invariant under rotations with respect to , it follows that the function is invariant under rotations, due to Proposition 10, we have that is radial, and therefore is well defined.
As we have seen,
Again, choosing , we have that is a contraction, and by Banach’s fixed point theorem, we have a unique that is solution to the nonlinear equation .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work has been partially supported by Project MECESUP2 PUC0711 and FONDECYT Grant no. 1130554.