Existence and stability of traveling waves for doubly degenerate diffusion equations

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 74; no. 2
• Main Authors: Huang, Rui, Liang, Zhanghua, Wang, Zhuangzhuang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2023; Springer Nature B.V
• Subjects: Diffusion rate; Energy methods; Engineering; Mathematical analysis; Mathematical Methods in Physics; Stability; Theoretical and Applied Mechanics; Traveling waves; Stability; 35B35; Doubly degenerate; Traveling waves; 35K59; 35K57; Existence; 35C07
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-023-01938-6

This paper is concerned with the existence and stability of traveling waves for doubly degenerate diffusion equations, where the spatial diffusion operator is of the form ∂ x ( | ∂ x u m | p - 2 ∂ x u m ) with m > 0 and p > 1 . It is proved that, for the slow diffusion case m ( p - 1 ) > 1 , there exists a minimum wave speed c ∗ , such that the problem admits smooth traveling waves when wave speed c > c ∗ and semi-finite traveling waves with critical wave speed c = c ∗ while, for the fast diffusion case 0 < m ( p - 1 ) < 1 , there is no nonnegative traveling wave solution. By the weighted energy method, we also show the L 1 -stability of the traveling waves.