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.