Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Abstract

This paper reports results on the classification of traveling wave solutions, including nonnegative traveling wave solutions in a weak sense, in the spatial 1D degenerate parabolic equation. These are obtained through dynamical systems theory and geometric approaches (in particular, Poincaré compactification). Classification of traveling wave solutions means enumerating those that exist and presenting properties of each solution, such as its profile and asymptotic behavior. The results examine a different range of parameters included in the equation, using the same techniques as discussed in the earlier work [Y. Ichida, Discrete Contin. Dyn. Syst., Ser. B, 28 (2023), no. 2, 1116-1132. In a clear departure from this previous work, the classification results obtained in this paper and the successful application of the known transformation also yield results for the classification of (weak) nonnegative traveling wave solutions for spatial 1D porous medium equations with special nonlinear terms. Finally, the bifurcations at infinity occur in the two-dimensional ordinary differential equations that characterize these traveling wave solutions are shown.