Asymptotic Stability of Singular Traveling Waves to Degenerate Advection-Diffusion Equations Under Small Perturbation

The main equation of this paper is the special case of equation studied by Il’in and Oleinik for single model equation with convex nonlinearity [ 4 ], by considering f ( u ) = u m and nonlinear diffusion, for m > 0 . We are interested in the stability of the degenerate advection-diffusion equatio...

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Bibliographic Details
Published inDifferential equations and dynamical systems Vol. 32; no. 4; pp. 965 - 977
Main Author Ghani, Mohammad
Format Journal Article
LanguageEnglish
Published New Delhi Springer India 01.10.2024
Springer Nature B.V
Subjects
Advection
Advection-diffusion equation
Computer Science
Engineering
Estimates
Mathematics
Mathematics and Statistics
Nonlinearity
Original Research
Perturbation
Stability
Traveling waves
Small perturbation
Stability
Degenerate advection-diffusion
35B40
35A01
Large wave amplitude
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Summary:The main equation of this paper is the special case of equation studied by Il’in and Oleinik for single model equation with convex nonlinearity [ 4 ], by considering f ( u ) = u m and nonlinear diffusion, for m > 0 . We are interested in the stability of the degenerate advection-diffusion equation by dealing with the singular term when u + = 0 . We first transform the original equation into the traveling waves by using the ansatz transformation. The weighted energy estimates of the transformed equation are then established, where the aim of this weighted function is to avoid the singular term when u + = 0 . At the final stage, the stability of traveling waves is shown based on the weighted energy estimates and appropriate perturbations.
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ISSN:0971-3514
0974-6870
DOI:10.1007/s12591-022-00602-1