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Abstract: We establish the existence of pulsating type entire solutions of reaction–advection–diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time . By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as in terms of appropriate subsolutions and supersolutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with traveling wave solutions of the equation where the dispersion kernel is nonnegative and the nonlinearity is monostable type. We show that there exists such that for any , there is a nonincreasing traveling wave solution with and , where is the smallest positive solution to . Furthermore, the existence of traveling wave solutions with is also established. For , we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense. [Copyright &y& Elsevier]

Abstract: We consider transition fronts (generalized traveling fronts) of mono-stable reaction–diffusion equations with spatially inhomogeneous nonlinearity. By constructing a cutoff function and using an approximate method, we establish the existence of transition fronts of the equation. Furthermore, we give the uniform non-degeneracy estimates of the solutions, such as a lower bound on the time derivative on some level sets, as well as an upper bound on the spatial derivative. [Copyright &y& Elsevier]