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

Yu Ichida; Takashi Okuda Sakamoto

doi:10.3934/dcds.2024030

Article Contents

2024, Volume 44, Issue 8: 2342-2367. Doi: 10.3934/dcds.2024030

This issue Previous Article Minimal $ P $-symmetric periodic solutions in semi-positive definite Hamiltonian systems Next Article On semidiscrete models dominated by the heat, wave and Laplace equations

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

Yu Ichida^1, , and
Takashi Okuda Sakamoto^2,

1.
Meiji University Organization for Strategic, Coordination of Research and Intellectual Properties, 1-1 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8301, Japan
2.
Graduate School of Science and Technology, Meiji University, 1-1-1, Higashimita Tama-ku Kawasaki Kanagawa 214-8571, Japan

^*Corresponding author: Yu Ichida
^*Corresponding author: Yu Ichida

Received: May 2023

Revised: March 2024

Early access: March 2024

Published: August 2024

IY was partially supported by JSPS KAKENHI Grant Number JP21J20035 and JP22KJ2844

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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.

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Mathematics Subject Classification: Primary: 35K65, 35C07, 34C05; Secondary: 35B40.

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Figure 1. Schematic pictures of the traveling wave solutions obtained in Theorem 2.8. Here it should be noted that the position of the singularity points $ \xi_{-} $ and $ \xi_{+} $ are not determined in our studies, however, they are shown in the figures for convenience. [Top left: The weak traveling wave solution in Theorem 2.8 (i).] [Top right: The weak traveling wave solution in Theorem 2.8 (ii).] [Lower center: The weak traveling wave solution in Theorem 2.8 (iii).]

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Figure 2. Schematic pictures of the traveling wave solutions obtained in Theorem 2.9. Here it should be noted that the position of the singularity point $ \xi_{-} $ is not determined in our studies, however, they are shown in the figures for convenience. [Left: The weak traveling wave solution in Theorem 2.9 (iv) in the case that $ D<0 $.] [Right: The traveling wave solution on $ \xi\in \mathbb{R} $ obtained in Theorem 2.9 (v) in the case that $ D>0 $.]

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Figure 3. Schematic pictures of the dynamics on the Poincaré disk in the case that $ 0<p<1 $. [Left: $ \delta = 0 $.] [Right: $ \delta = 1 $.]

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Figure 4. Schematic diagram of the finite equilibria and equilibria at infinity on the Poincaré disk. White and Black circles denote source and sink equilibria, respectively. Gray squares denote saddle equilibria, and gray diamonds equilibria with the center manifold. Left, center and right disks correspond to the case that $ 0<p<1 $, $ p = 1 $ and $ p>1 $, respectively

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Figure 5. Schematic picture of the bifurcation on $ \overline{U}_2 $. The gray circles correspond to the equilibria $ E_3 $ and $ E_4 $ on $ \overline{U}_2 $

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Figure 6. Locations of the Poincaré sphere and chart $ \overline{U}_{2} $

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