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 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 H？rmander 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