Traveling wave behavior for a nonlinear reaction-diffusion equation

There is the widespread existence of wave phenomena in physics, chemistry and biology. In the present paper, we study a nonlinear reaction-diffusion equation, which can be regarded as a generalized Fisher equation. Applying the bifurcation theory of planar systems, bifurcations of bell-profile waves and kink-profile waves for the generalized Fisher equation are illustrated under certain parameter conditions. From there, a bounded traveling wave solution is obtained by means of a series of nonlinear coordinate transformations. At the end of the paper, the asymptotic behaviors of proper solutions for the generalized Fisher equation are established by applying the qualitative theory of differential equations.