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Stability of solutions to contact Hamilton-Jacobi equation on the circle

Yang Xu, Jun Yan, Kai Zhao

Abstract

Combing the weak KAM method for contact Hamiltonian systems and the theory of viscosity solutions for Hamilton-Jacobi equations, we study the Lyapunov stability and instability of viscosity solutions for evolutionary contact Hamilton-Jacobi equation in the first part. In the second part, we study the existence and multiplicity of time-periodic solutions.

Stability of solutions to contact Hamilton-Jacobi equation on the circle

Abstract

Combing the weak KAM method for contact Hamiltonian systems and the theory of viscosity solutions for Hamilton-Jacobi equations, we study the Lyapunov stability and instability of viscosity solutions for evolutionary contact Hamilton-Jacobi equation in the first part. In the second part, we study the existence and multiplicity of time-periodic solutions.
Paper Structure (10 sections, 14 theorems, 89 equations)

This paper contains 10 sections, 14 theorems, 89 equations.

Table of Contents

  1. Introduction and main results
  2. Assumptions and the motivation of this paper
  3. Main results
  4. Construction of subsolutions and supersolutions
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['thm1']} (1)
  7. Proof of Theorem \ref{['thm1']} (2)
  8. Proof of Theorem \ref{['thm2']}
  9. Examples
  10. Appendix

Key Result

Theorem 1.2

Assume (H1)-(H3) and A. Set then there exists $\delta_0\in \mathbb{R}^+$ such that

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 19 more