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Globally solvable time-periodic evolution equations in Gelfand-Shilov classes

Fernando de Ávila Silva, Marco Cappiello

Abstract

In this paper we consider a class of evolution operators with coefficients depending on time and space variables $(t,x) \in \mathbb{T} \times \mathbb{R}^n$, where $\mathbb{T}$ is the one-dimensional torus and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand-Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on $\mathbb{R}^n$.

Globally solvable time-periodic evolution equations in Gelfand-Shilov classes

Abstract

In this paper we consider a class of evolution operators with coefficients depending on time and space variables , where is the one-dimensional torus and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand-Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on .
Paper Structure (11 sections, 20 theorems, 177 equations)

This paper contains 11 sections, 20 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Time-periodic Gelfand-Shilov spaces and eigenfunction expansions
  4. Discretization of equation $Lu=f$ and global hypoellipticity
  5. Global solvability
  6. Time independent coefficients
  7. Application: Cauchy problem
  8. Reduction to the normal form
  9. Proof of the main theorem
  10. Sufficient conditions
  11. Necessary conditions

Key Result

Theorem 1

Let $L$ be defined by op-intro, P-intro, with $P$ self-adjoint and satisfying P-elliptic. Then:

Theorems & Definitions (40)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • ...and 30 more