Entire solutions in nonlocal monostable equations: Asymmetric case

Yu-Juan Sun; Li Zhang; Wan-Tong Li; Zhi-Cheng Wang

doi:10.3934/cpaa.2019051

Article Contents

2019, Volume 18, Issue 3: 1049-1072. Doi: 10.3934/cpaa.2019051

This issue Previous Article Uniqueness for Neumann problems for nonlinear elliptic equations Next Article The properties of positive solutions to semilinear equations involving the fractional Laplacian

Entire solutions in nonlocal monostable equations: Asymmetric case

1.
State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, Shaanxi 710071, China
2.
School of Science, Chang'an University, Xi'an, Shaanxi 710064, China
3.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author
* Corresponding author

Received: October 31, 2017

Revised: July 31, 2018

Published: May 2019

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Abstract

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., $u_{t}=J*u-u+f(u)$. Here the kernel $J$ is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds $c^{*}$ and $\hat{c}^{*}$, where $c^*$ and $\hat{c}^{*}$ are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel $J$ is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

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Mathematics Subject Classification: Primary: 35K57, 35B08; Secondary: 34C14.

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Table 1. Region of $(c, \hat{c})$

	$\hat{c}^{*}>0$	$\hat{c}^{*}=0$	$\hat{c}^{*} <0$
${c^{\ast}>0}$	$C_{11}$	$C_{12}$	$ C_{13}=C^{1}_{13}\cup C^{2}_{13} $
${c^{\ast}=0}$	$C_{21}$	$C_{22}$	$C_{23}=C^{1}_{23}\cup C^{2}_{23} $
${c^{\ast} <0}$	$C_{31}=C^{1}_{31}\cup C^{2}_{31} $	$C_{32}=C^{1}_{32}\cup C^{2}_{32}$	$\setminus$

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