Semilinear pseudodifferential equations in spaces of tempered ultradistributions

Abstract

We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a sub-exponential growth of its symbol and that the non linear part is given by an infinite sum of powers of u with sub-exponential growth with respect to u, we prove a regularity result in the functional setting of the quoted ultradistribution spaces for a weak Sobolev type solution u. * Corresponding author: marco.cappiello@unito.it order, is used also here but with a more advanced technique since we consider infinite series both in the case of symbols and in the case of nonlinear terms. Examples of our elliptic symbols are a(x, ξ) = e c (x,ξ) 1/m , m > 1, c ∈ R, whereas for what concerns the nonlinear terms we can consider F [u] = β c β P β (x)u |β| where P β (x) are ultrapolynomials of the form γc γ x γ /γ! m , m > 1 and c β are suitable complex numbers tending rapidly to zero. Actually, we will consider nonlinear terms also with the additional assumption that u ∈ H s (R d ), s > d/2. With this we have plenty of examples, for example F [u] = P (x) cos u or F [u] = P (x)e u k , k ∈ Z + , where P has sub-exponential growth of the order related to the order of the growth of the symbol. In this way we can analyse elliptic operators A of infinite order and sub-exponential growth as well as nonlinear terms with sub-exponential growth, not considered in the literature, which shows an intrinsic connection of the pseudodifferential calculus of [23] with the spaces of ultradistributions.The paper is organised as follows. In the next Section 1 we introduce the main tools involved in the paper and state the main result, namely Theorem 1.2. Section 2 contains some examples of elliptic operators and nonlinear terms which motivate our analysis and for which Theorem 1.2 holds. In Section 3 we refine some results about the pseudodifferential operators studied in [23] and we prove some precise estimates for the norms of some composed operators which will be instrumental in the proof of Theorem 1.2. Finally, Section 4 is devoted to the proof of the theorem which will be divided in two parts, one corresponding to the proof of the decay properties of the solution and the other related to its regularity. Notation and the main theoremBefore stating our results, let us fix some notation and introduce the functional setting where they are obtained. In the sequel, the sets of integer, non-negative integer, positive integer, real and complex numbers are denoted as standard by Z, N, Z + , R, C. We denote x = (1 + |x| 2 ) 1/2 forFixed B > 0, we shall denote by Q c B the set of all (x, ξ) ∈ R 2d for which we have x ≥ B or ξ ≥ B. Finally, for s ∈ R, we shall denote by H s (R d ) the Sobolev space of all u ∈ S ′ (R d ) for which ξ sû (ξ) ∈ L 2 (R d ), whereû denotes the Fourier transform of u. Following [14], in the sequel we shall consider sequences M p of positive numbers such that M 0 = M 1 = 1 and satisfying...

Spectral asymptotics for infinite order pseudo-differential operators

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Infinite order $$\Psi $$DOs: composition with entire functions, new Shubin-Sobolev spaces, and index theorem

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Pseudo-Differential Calculus in Anisotropic Gelfand–Shilov Setting

“…After their appearance, Gelfand-Shilov spaces have been recognized as a natural functional setting for pseudo-differential and Fourier integral operators, due to their nice behavior under Fourier transformation, and applied in the study of several classes of partial differential equations, see e. g. [1,[3][4][5][6][7][8].…”

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