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

  • * Corresponding author: Yu Ichida

    * Corresponding author: Yu Ichida

The author was partially supported by JSPS KAKENHI Grant Number JP21J20035

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  • 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.

    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 $.]

    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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