Analysis of traveling fronts for chemotaxis model with the nonlinear degenerate viscosity

Mohammad Ghani; Mohammad Ghani

doi:10.3934/math.20231527

AIMS Mathematics

2023, Volume 8, Issue 12: 29872-29891. doi: 10.3934/math.20231527

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Research article

Analysis of traveling fronts for chemotaxis model with the nonlinear degenerate viscosity

Mohammad Ghani ^,

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received: 19 August 2023 Revised: 30 October 2023 Accepted: 01 November 2023 Published: 03 November 2023
MSC : 35A01, 35B40, 35Q92, 92C17

In this paper, we are interested in chemotaxis model with nonlinear degenerate viscosity under the assumptions of $ \beta = 0 $ (without the effect of growth rate) and $ u_+ = 0 $. We need the weighted function defined in Remark 1 to handle the singularity problem. The higher-order terms of this paper are significant due to the nonlinear degenerate viscosity. Therefore, the following higher-order estimate is introduced to handle the energy estimate:

$ \begin{equation*} \begin{split} &U^{m-2} = \left( \frac{1}{U} \right)^{2-m}\leq Kw(z)\leq \frac{Cw(z)}{U}, \;\text{if}\;0<m<2, \\ &U^{m-2}\leq Lu_-\leq\frac{Cu_-}{U}, \;\text{if}\;m\geq 2, \end{split} \end{equation*} $

where $ C = max\left\{ K, L \right\} = max\left\{ \frac{a}{m-a}, (m+a)^m \right\} $ for $ a > 0 $ and $ m > a $, and $ w(z) $ is the weighted function. Then we show that the traveling waves are stable under the appropriate perturbations. The proof is based on a Cole-Hopf transformation and weighted energy estimates.
- chemotaxis model,
- nonlinear diffusion,
- stability,
- existence,
- weighted energy estimates
Citation: Mohammad Ghani. Analysis of traveling fronts for chemotaxis model with the nonlinear degenerate viscosity[J]. AIMS Mathematics, 2023, 8(12): 29872-29891. doi: 10.3934/math.20231527

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Abstract

In this paper, we are interested in chemotaxis model with nonlinear degenerate viscosity under the assumptions of $ \beta = 0 $ (without the effect of growth rate) and $ u_+ = 0 $. We need the weighted function defined in Remark 1 to handle the singularity problem. The higher-order terms of this paper are significant due to the nonlinear degenerate viscosity. Therefore, the following higher-order estimate is introduced to handle the energy estimate:

$ \begin{equation*} \begin{split} &U^{m-2} = \left( \frac{1}{U} \right)^{2-m}\leq Kw(z)\leq \frac{Cw(z)}{U}, \;\text{if}\;0<m<2, \\ &U^{m-2}\leq Lu_-\leq\frac{Cu_-}{U}, \;\text{if}\;m\geq 2, \end{split} \end{equation*} $

where $ C = max\left\{ K, L \right\} = max\left\{ \frac{a}{m-a}, (m+a)^m \right\} $ for $ a > 0 $ and $ m > a $, and $ w(z) $ is the weighted function. Then we show that the traveling waves are stable under the appropriate perturbations. The proof is based on a Cole-Hopf transformation and weighted energy estimates.

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