Transition waves for Fisher–KPP equations with general time-heterogeneous and space-periodic coefficients

Abstract

This paper is devoted to existence and non-existence results for generalized\ntran-sition waves solutions of space-time heterogeneous Fisher-KPP equations.\nWhen the coefficients of the equation are periodic in space but otherwise\ndepend in a fairly gen-eral fashion on time, we prove that such waves exist as\nsoon as their speed is sufficiently large in a sense. When this speed is too\nsmall, transition waves do not exist anymore, this result holds without\nassuming periodicity in space. These necessary and sufficient conditions are\nproved to be optimal when the coefficients are periodic both in space and time.\nOur method is quite robust and extends to general non-periodic space-time\nheterogeneous coefficients, showing that transition waves solutions of the\nnonlinear equation exist as soon as one can construct appropriate solutions of\na given linearized equation.\n