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The Wave Equation on Lattices and Oscillatory Integrals

Cheng Bi, Jiawei Cheng, Bobo Hua

Abstract

In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove that the optimal time decay rate of the fundamental solution is of order $|t|^{-\frac{3}{2}}\log |t|$, which is the first extension of P. Schultz's results \cite{S98} in $d=2,3$ to the higher dimension. Moreover, we notice that the Newton polyhedron can be used not only to interpret the decay rates for $d=2,3,4$, but also to study the most degenerate case for all odd $d\geq 3$. Furthermore, we prove $l^p\rightarrow l^q$ estimates as well as Strichartz estimates and give applications to nonlinear wave equations.

The Wave Equation on Lattices and Oscillatory Integrals

Abstract

In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice with dimension . Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove that the optimal time decay rate of the fundamental solution is of order , which is the first extension of P. Schultz's results \cite{S98} in to the higher dimension. Moreover, we notice that the Newton polyhedron can be used not only to interpret the decay rates for , but also to study the most degenerate case for all odd . Furthermore, we prove estimates as well as Strichartz estimates and give applications to nonlinear wave equations.
Paper Structure (20 sections, 26 theorems, 167 equations, 2 tables)

This paper contains 20 sections, 26 theorems, 167 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics on $l^p({\mathbb Z}^d)$ and the DW
  4. Results on uniform estimates.
  5. Proof of Theorem \ref{['main-thm']}
  6. Classification of the critical points
  7. Reductions
  8. The most degenerate case
  9. Proof of Proposition \ref{['thm-pro']}
  10. $(v_0,\xi_0)\in\Omega_3\times\Sigma_3$
  11. $(v_0,\xi_0)\in(\Omega_2\backslash\Omega_1)\times\Sigma_2$
  12. $(v_0,\xi_0)\in\Omega_1\times\Sigma_1$
  13. Newton Polyhedra
  14. Basic concepts and results
  15. Application to the DW
  16. ...and 5 more sections

Key Result

Theorem 1.1

There exists $C>0$ such that

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 38 more