Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

Abstract

This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in Liang and Zhao (2010 J. Funct. Anal. 259 857–903), Nadin (2009 J. Math. Pures Appl. 92 232–62), Nolen et al (2005 Dyn. PDE 2 1–24), Nolen and Xin (2005 Discrete Contin. Dyn. Syst. 13 1217–34) and Weinberger (2002 J. Math. Biol. 45 511–48) for random dispersal Fisher-KPP equations with time and space periodic dependence, in Nadin and Rossi (2012 J. Math. Pures Appl. 98 633–53), Nadin and Rossi (2015 Anal. PDE 8 1351–77), Nadin and Rossi (2017 Arch. Ration. Mech. Anal. 223 1239–67), Shen (2010 Trans. Am. Math. Soc. 362 5125–68), Shen (2011 J. Dynam. Differ. Equ. 23 1–44), Shen (2011 J. Appl. Anal. Comput. 1 69–93), Tao et al (2014 Nonlinearity 27 2409–16) and Zlatoš (2012 J. Math. Pures Appl. 98 89–102) for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in Coville et al (2013 Ann. Inst. Henri Poincare 30 179–223), Rawal et al (2015 Discrete Contin. Dyn. Syst. 35 1609–40) and Shen and Zhang (2012 Comm. Appl. Nonlinear Anal. 19 73–101) for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable for well-fitted perturbation. Up to the author’s knowledge, it is the first time that the stability of transition waves of Fisher-KPP equations with general time and space dependence is studied.