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Curved fronts of combustion reaction-diffusion equations

Wei-Jie Sheng, Xin-Tian Zhang

Abstract

This paper is concerned with curved fronts of combustion reaction-diffusion equations in $\mathbb{R}^N$ $(N\geq2)$. By mixing finite planar fronts and constructing suitable super- and subsolutions, we prove the existence, uniqueness and stability of polytope-like curved fronts in $\mathbb{R}^N$. Besides, we show that these curved fronts are transition fronts.

Curved fronts of combustion reaction-diffusion equations

Abstract

This paper is concerned with curved fronts of combustion reaction-diffusion equations in . By mixing finite planar fronts and constructing suitable super- and subsolutions, we prove the existence, uniqueness and stability of polytope-like curved fronts in . Besides, we show that these curved fronts are transition fronts.
Paper Structure (7 sections, 13 theorems, 252 equations)

This paper contains 7 sections, 13 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Existence and uniqueness
  4. A hypersurface with asymptotic planes
  5. Construction of supersolutions
  6. Proofs of Theorems \ref{['Theorem 2.1']} and \ref{['Theorem 2.3']}
  7. Stability of curved fronts

Key Result

Proposition 1.1

Under the assumptions $(F1)$ and $(F2)$, there exist positive constants $L_{1}$, $L_{2}$, $L_{3}$, $L_{4}$ and $\beta_{0}$ such that and Moreover, there exists a positive constant $\gamma_{\star}$ such that $\gamma_{\star}\leq \min \{\theta / 4,(1-\theta)/2, \sigma / 4\}$ and

Theorems & Definitions (15)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 2.1: Existence
  • Remark 2.2
  • Theorem 2.3: Uniqueness
  • Theorem 2.4: Stability
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 5 more