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Entire solutions of the Fisher–KPP equation on the half line

Part of: Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  26 March 2019

BENDONG LOU ,
JUNFAN LU  and
YOSHIHISA MORITA
BENDONG LOU
Affiliation:
Department of Mathematics, Mathematics & Science College, Shanghai Normal University, Shanghai200234, China email: lou@shnu.edu.cn
JUNFAN LU
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai200092, China email: 1410541@tongji.edu.cn
YOSHIHISA MORITA
Affiliation:
Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu520-2194, Japan email: morita@rins.ryukoku.ac.jp
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Abstract

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

Keywords

Reaction–diffusion equationFisher–KPP equationentire solutiontraveling wave solution

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B08: Entire solutions 35B40: Asymptotic behavior of solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 3 , June 2020 , pp. 407 - 422
Copyright
© Cambridge University Press 2019

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Footnotes

†

This research was partly supported by the NSFC (No. 11671262) and JSPS KAKENHI Grant Number, 18H01139, 26247013 and JST CREST Grant Number JPMJCR14D3.

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