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The Cauchy problem for $p$-evolution equations with variable coefficients in Gelfand-Shilov spaces

Marco Cappiello, Eliakim Cleyton Machado

TL;DR

The paper addresses the Cauchy problem for linear $p$-evolution operators with time-dependent and spatially decaying lower-order coefficients, establishing well-posedness in Gelfand-Shilov spaces $\mathcal S_s^\theta(\mathbb R)$ by reducing to Gevrey spaces through a conjugation by $e^{\delta\langle x\rangle^{1/s}}$. The main technique combines a detailed conjugation analysis that preserves the principal part while adding controlled decays to the lower-order terms, with a Gevrey-well-posedness framework in weighted Gevrey-Sobolev spaces; this yields existence, uniqueness, and energy estimates in GS spaces under the threshold $(p-1)\theta<\min\{1/(1-\sigma),s\}$. The paper also proves ill-posedness results for model operators when the threshold is violated, demonstrating the sharpness of the well-posedness condition in many cases. Overall, the work extends Gevrey-well-posedness results to Gelfand-Shilov spaces for high-order evolution equations with variable coefficients and clarifies the critical balance between coefficient decay, operator order, and function-space regularity. These findings provide a rigorous analytical framework for the Cauchy problem in GS spaces with variable coefficients, relevant to dispersive and Schrödinger-type models.

Abstract

We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $|x|$ large, we prove a well-posedness result in Gelfand-Shilov spaces.

The Cauchy problem for $p$-evolution equations with variable coefficients in Gelfand-Shilov spaces

TL;DR

The paper addresses the Cauchy problem for linear -evolution operators with time-dependent and spatially decaying lower-order coefficients, establishing well-posedness in Gelfand-Shilov spaces by reducing to Gevrey spaces through a conjugation by . The main technique combines a detailed conjugation analysis that preserves the principal part while adding controlled decays to the lower-order terms, with a Gevrey-well-posedness framework in weighted Gevrey-Sobolev spaces; this yields existence, uniqueness, and energy estimates in GS spaces under the threshold . The paper also proves ill-posedness results for model operators when the threshold is violated, demonstrating the sharpness of the well-posedness condition in many cases. Overall, the work extends Gevrey-well-posedness results to Gelfand-Shilov spaces for high-order evolution equations with variable coefficients and clarifies the critical balance between coefficient decay, operator order, and function-space regularity. These findings provide a rigorous analytical framework for the Cauchy problem in GS spaces with variable coefficients, relevant to dispersive and Schrödinger-type models.

Abstract

We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for large, we prove a well-posedness result in Gelfand-Shilov spaces.
Paper Structure (8 sections, 12 theorems, 125 equations)

This paper contains 8 sections, 12 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Background theory and idea of the proof
  3. Conjugation of $P$ by $e^{\delta\langle x\rangle^{1/s}}$
  4. Proof of Theorem \ref{['theorem_main_result_3']}
  5. Ill-posedness results for model operators
  6. Proof of Proposition \ref{['proposition_main_result_4']}
  7. Proof of Lemma \ref{['proposition_estimates_E_k_below']}
  8. Pseudodifferential operators

Key Result

Theorem 1

Let $\theta_0>1$ and $\sigma \in \left( \frac{p-2}{p-1},1 \right)$ such that $\theta_0 < \frac{1}{(p-1)(1-\sigma)}$. Let $P$ be an operator of the type differential_p_evolution_operator whose coefficients satisfy the following assumptions: Let $s,\theta>1$ such that $(p-1)\theta < \min \left\lbrace \frac{1}{1-\sigma}, s \right\rbrace$ and $\theta \geq \theta_0$, and let $f \in C \left( [0,T];H_{\

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • Remark 2
  • Lemma 1
  • proof
  • ...and 13 more