Existence and stability of generalized transition waves for time heterogeneous reaction-diffusion equations in a cylinder

The current paper is devoted to the study of existence, stability and propagation speed of generalized transition waves for time heterogeneous reaction-diffusion equations in a cylinder. Here the reaction term is of Fisher-KPP type and depends in a general way on t∈R. As far as general time heterogeneity and cylindrical domain are concerned, the generalized principal eigenvalue of a linear parabolic operator with Neumann boundary condition is used to address the difficulty associated with the principal eigenvalue problem. We first investigate the existence and nonexistence of generalized transition waves for such equations. Especially, we establish the existence of critical generalized transition waves under the assumption on a time mean value, by constructing suitable sub- and supersolutions in the critical case. Then the spreading properties for solutions of the corresponding Cauchy problem with compactly supported initial values are proven. Finally, by proper modifications of the squeezing technique, we also show the stability of generalized transition waves.