Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions

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• 作者：Karel Hasik^a ; Jana Kopfová ; ^a ; Petra Ná ; bělková ; ^a ; Sergei Trofimchuk^b ; ^{trofimch@inst-mat.utalca.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34K12 ; 35K57 ; 92D25
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：6130-6175
• 全文大小：622 K

文摘

We consider the nonlocal KPP-Fisher equation u_t(t,x)=u_xx(t,x)+u(t,x)(1−(K⁎u)(t,x))$u_t(t,x)=u_{xx}(t,x)+u(t,x)(1-(K⁎u)(t,x))$ which describes the evolution of population density u(t,x)$u(t,x)$ with respect to time t and location x  . The non-locality is expressed in terms of the convolution of u(t,⋅)$u(t,\cdot )$ with kernel 13d2000f828cbc27469870d2" title="Click to view the MathML source">K(⋅)≥0$K(\cdot )\ge 0$, ∫_RK(s)ds=1${\int }_{\mathbb{R}}K(s)ds=1$. The restrictions d99b55852b628cf152884083a48" title="Click to view the MathML source">K(s)$K(s)$, s≥0$s\ge 0$, and d99b55852b628cf152884083a48" title="Click to view the MathML source">K(s)$K(s)$, s≤0$s\le 0$, are responsible for interactions of an individual with his left and right neighbors, respectively. We show that these two parts of K play quite different roles as for the existence and uniqueness of traveling fronts to the KPP-Fisher equation. In particular, if the left interaction is dominant, the uniqueness of fronts can be proved, while the dominance of the right interaction can induce the co-existence of monotone and oscillating fronts. We also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on K.