# Postdoc Seminar

## Introduction to Entire Solutions of the KPP Equation, I

### 王丽娜 博士 (华师大PDE中心博士后)

#### 2012年2月21日(周二)下午 1:00-3:00 行政楼12楼偏微分方程中心1202报告厅

数学系-偏微分方程中心联合讨论班

摘 要： In this talk, I would like to introduce F. Hamel and N. Nadirashvili’s results on entire solutions of the KPP equation, which is published on “Comm. Pure Appl. Math”. The authors deal with the solutions defined for all time of the KPP equation ${{u}_{t}}={{u}_{xx}}+f(u)$, $0<$$u(x,t)<1$, $(x,t)\in {{\mathbb{R}}^{2}}$, where $f$ is a KPP-type nonlinearity defined in $[0,1]$: $f(0)=f(1)=0$, ${{f}^{'}}(0)>0$, ${{f}^{'}}(1)<0$, $f>0$ in $(0,1)$, and ${{f}^{'}}(s)\le {{f}^{'}}(0)$ in $[0,1]$. This equation admits infinitely many traveling-wave-type solutions, increasing or decreasing in $x$. It also admits solutions that depend only on $t$. In this paper, the authors build four other manifolds of solutions: One is 5-dimensional, one is 4-dimensional, and two are 3-dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling wave solutions are on the boundary of these four manifolds.