Existence of pulsating waves of advection–reaction–diffusion equations of ignition type by a new method Original Research Article

The goal of this paper is to study the advection–reaction–diffusion equation of ignition type: ut−Δu+q(x,y)⋅∇u=f(u).ut−Δu+q(x,y)⋅∇u=f(u). Turn MathJax on Xin [J.X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal. 121 (1992) 205–233], Berestycki and Hamel [H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002) 949–1032] proved the existence of pulsating waves, using the theory of degenerate elliptic equations. Our goal is to give an alternative and rather natural proof of the same result: first we transform the problem so that the existence of pulsating waves is equivalent to the existence of 1-periodic in time solutions of a nonlinear parabolic equation with 1-periodic in time coefficients; next we prove the existence of such periodic solutions, using a continuation method, based on the implicit function theorem.