Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

Yu Ichida

doi:10.3934/dcdsb.2022114

Article Contents

2023, Volume 28, Issue 2: 1116-1132. Doi: 10.3934/dcdsb.2022114

This issue Previous Article Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises Next Article Global attractor for the periodic generalized Korteweg-De Vries equation through smoothing

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

Yu Ichida^,

Graduate School of Science and Technology, Meiji University / JSPS Research Fellow, 1-1-1, Higashimita Tama-ku Kawasaki Kanagawa 214-8571, Japan

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

Received: September 30, 2021

Revised: February 28, 2022

Early access: June 2022

Published: February 2023

The author was partially supported by JSPS KAKENHI Grant Number JP21J20035

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Abstract

Traveling wave solutions for the one-dimensional degenerate parabolic equations are considered. The purpose of this paper is to classify the nonnegative traveling wave solutions including sense of weak solutions of these equations and to present their existence, information about their shape and asymptotic behavior. These are studied by applying the framework that combines Poincaré compactification and classical dynamical systems theory. We also aim to use these results to generalize the results of our previous studies. The key to this is the introduction of a transformation, which overcomes the generalization difficulties faced by these studies.

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

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Figure 1. Schematic picture of the traveling wave solutions obtained in Theorems. 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 the convenience. [Top left: The weak traveling wave solution with singularity in Theorem 2.3.] [Top right: The weak traveling wave solution with singularity in Theorem 2.4 in the case that $ D<0 $.] [Lower center: The traveling wave solution on $ \xi\in \mathbb{R} $ obtained in Theorem 2.5 in the case that $ D>0 $.]

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Figure 2. Schematic pictures of the dynamics on the Poincaré disk in the case that $ \delta = 0 $ or $ 1 $, $ 1<p\in \mathbb{R} $. [Left: Case $ \delta = 0 $.] [Right: Case $ \delta = 1 $.]

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