Geometric Structure of the Traveling Waves for 1D Degenerate Parabolic Equation

We clarify the geometric structure of non-negative traveling waves for the spatial one-dimensional degenerate parabolic equation \(U_{t}=U^{p}(U_{xx}+\mu U)-\delta U\). This equation has a nonlinear term with a parameter \(p>0\) and the cases \(0<p<1\) and \(p>1\) have been investigated in the author’s previous studies. It has been pointed out that the classifications of the traveling waves for these two cases are not the same and thus a bifurcation phenomenon occurs at \(p=1\). However, the classification of the case \(p=1\) remains open since the conventional approaches do not work for this case, which have prevented us to understand how the traveling waves bifurcate. The difficulty for the case \(p=1\) is that the corresponding ordinary differential equation through the Poincaré compactification has the non-hyperbolic equilibrium at infinity and we need to estimate the asymptotic behaviors of the trajectories near it. In this paper, we solve this problem by using a new asymptotic approach, which is completely different from the asymptotic analysis performed in the previous studies, and clarify the structure of the traveling waves in the case of \(p=1\). We then discuss the rich structure of traveling waves of the equation from a geometric point of view.