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Gevrey well posedness for $3$-evolution equations with variable coefficients

Alexandre Arias Junior, Alessia Ascanelli, Marco Cappiello

Abstract

We study the Cauchy problem for a class of third order linear anisotropic evolution equations with complex valued lower order terms depending both on time and space variables. Under suitable decay assumptions for $|x| \to \infty$ on these coefficients, we prove a well posedness result in Gevrey-type spaces.

Gevrey well posedness for $3$-evolution equations with variable coefficients

Abstract

We study the Cauchy problem for a class of third order linear anisotropic evolution equations with complex valued lower order terms depending both on time and space variables. Under suitable decay assumptions for on these coefficients, we prove a well posedness result in Gevrey-type spaces.

Paper Structure

This paper contains 8 sections, 11 theorems, 120 equations.

Table of Contents

  1. Introduction and main result
  2. Pseudodifferential operators of infinite order
  3. Change of variables
  4. Sobolev well-posedness for the Cauchy problem \ref{['equation_conjugated_cauchy_problem']}
  5. Conjugation of $e^{\tilde{\Lambda}} (iP) \{e^{\tilde{\Lambda}}\}^{-1}$ by $e^{k(t)\langle D \rangle^{\frac{1}{\theta}}_{h}}$
  6. Lower bound estimates for the real parts
  7. Gevrey well-posedness for the Cauchy problem \ref{['Cauchy_problem_in_introduction']}
  8. Appendix

Key Result

Theorem 1

Let $s_0 > 1$ and $\sigma \in (\frac{1}{2}, 1)$ such that $s_0 < \frac{1}{2(1-\sigma)}$. Let moreover $P(t,x,D_t, D_x)$ be defined by diffP. Assume that Then given $\theta\in\left[s_0,\frac{1}{2(1-\sigma)}\right), \rho >0$, $m\in\mathbb R$, and given $f \in C([0,T], H^m_{\rho; \theta}(\mathbb R))$ and $g \in H^m_{\rho; \theta}(\mathbb R)$, there exists a unique solution $u \in C^1([0,T], H^{m}_{

Theorems & Definitions (32)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Remark 4
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 22 more