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

We study existence and nonexistence results for generalized transition wave solutions of space-time heterogeneous Fisher-KPP equations. When the coefficients of the equation are periodic in space but otherwise depend in a fairly general fashion on time, we prove that such waves exist as soon as their speed is sufficiently large in a sense. When this speed is too small, transition waves do not exist anymore; this result holds without assuming periodicity in space. These necessary and sufficient conditions are proved to be optimal when the coefficients are periodic both in space and time. Our method is quite robust and extends to general nonperiodic space-time heterogeneous coefficients, showing that transition wave solutions of the nonlinear equation exist as soon as one can construct appropriate solutions of a given linearized equation.