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The Cauchy problem for $p$-evolution equations with variable coefficients in Gevrey classes

Alexandre Arias Junior, Alessia Ascanelli, Marco Cappiello, Eliakim Cleyton Machado

TL;DR

This work proves Gevrey-type well-posedness for the Cauchy problem of linear $p$-evolution equations in one space dimension with coefficients depending on $(t,x)$. It develops a two-step infinite-order conjugation $Q_{oldsymbol{ ext{}}oldsymbol{ ext{}}}=e^{oldsymbol{ ext{}}igl(oldsymbol{ ext{}} igr)}(t,D)\nolimits e^oldsymbol{(}x,D)$ built from carefully designed phase functions $oldsymbol{ ext{}}igl(\lambda_{p-k}(x,\xi)igr)$ to absorb imaginary parts of lower-order terms, and a large damping multiplier $K(T-t)ra \xira_h^{(p-1)(1-\sigma)}$ to move the problem into a Sobolev setting. The authors exploit SG calculus, an explicit invertible construction for $e^{oldsymbol{ ext{}}igl(oldsymbol{ ext{}} igr)}$, and sharp Gårding inequalities to obtain lower bounds on the real parts of transformed symbols, enabling energy estimates. Under decay and regularity assumptions on the lower-order coefficients $a_{p-j}$, they prove the main result: existence and uniqueness of solutions in Gevrey-Sobolev spaces $H^m_{ ho; heta}$ with a controlled loss of the Gevrey weight, yielding well-posedness in $igcup_{ ho>0}H^m_{ ho; heta}$ for $ heta$ in a specified range. The approach extends earlier $p=2,3$ results to general $p$ in 1D and provides a robust framework for future higher-dimensional extensions.

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 Gevrey-type spaces.

The Cauchy problem for $p$-evolution equations with variable coefficients in Gevrey classes

TL;DR

This work proves Gevrey-type well-posedness for the Cauchy problem of linear -evolution equations in one space dimension with coefficients depending on . It develops a two-step infinite-order conjugation built from carefully designed phase functions to absorb imaginary parts of lower-order terms, and a large damping multiplier to move the problem into a Sobolev setting. The authors exploit SG calculus, an explicit invertible construction for , and sharp Gårding inequalities to obtain lower bounds on the real parts of transformed symbols, enabling energy estimates. Under decay and regularity assumptions on the lower-order coefficients , they prove the main result: existence and uniqueness of solutions in Gevrey-Sobolev spaces with a controlled loss of the Gevrey weight, yielding well-posedness in for in a specified range. The approach extends earlier results to general in 1D and provides a robust framework for future higher-dimensional extensions.

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 Gevrey-type spaces.
Paper Structure (10 sections, 10 theorems, 125 equations)

This paper contains 10 sections, 10 theorems, 125 equations.

Table of Contents

  1. Introduction and main results
  2. Pseudodifferential operators
  3. Construction of a change of variable
  4. The functions $\lambda_{p-k}(x,\xi)$, $k=1,...,p-1$
  5. Invertibility of the operator $e^\Lambda(x,D)$
  6. Conjugation of the operator $iP$
  7. Conjugation of $iP$ by $e^\Lambda$
  8. Conjugation of $e^\Lambda(iP)(e^\Lambda)^{-1}$ by $e^{\Lambda_{K,\rho^\prime}}$
  9. Estimates from below for the real parts
  10. Proof of Theorem \ref{['maintheorem1']}

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: If $\theta>1$ is such that $\theta_0 \leq \theta < \frac{1}{(p-1)(1-\sigma)}$, the data $f \in C\left([0,T];H_{\rho;\theta}^m(\mathbb R)\right)$ and $g \in H_{\r

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Lemma 1
  • proof
  • Remark 4
  • ...and 9 more