Stability of traveling waves of the Keller–Segel system with logarithmic sensitivity

Proceeding with a series of works (Refs. 12, 23–25) by the authors, this paper establishes the nonlinear asymptotic stability of traveling wave solutions of the Keller–Segel system with nonzero chemical diffusion and linear consumption rate, where the right asymptotic state of cell density is vacuum (zero) and the initial value is a perturbation with zero integral from the spatially shifted traveling wave. The main challenge of the problem is various singularities caused by the logarithmic sensitivity and the vacuum asymptotic state, which are overcome by a Hopf–Cole type transformation and the weighted energy estimates with an unbounded weight function introduced in the paper.

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CAUCHY PROBLEM FOR A MODEL SYSTEM OF THE RADIATING GAS: WEAK SOLUTIONS WITH A JUMP AND CLASSICAL SOLUTIONS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202599000063 ◽

1999 ◽

Vol 09 (01) ◽

pp. 69-91 ◽

Cited By ~ 40

Author(s):

SHUICHI KAWASHIMA ◽

SHINYA NISHIBATA

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Weak Solutions ◽

Traveling Wave ◽

Weak Sense ◽

Critical Value ◽

Classical Solutions ◽

Asymptotic State ◽

Spatial Derivative ◽

The Right

This paper deals with the global existence and the time asymptotic state of solutions to the initial value problems for the system derived from approximating a one-dimensional model of a radiating gas. When the spatial derivative of the initial data is larger than a certain negative critical value, a unique solution exists globally in time. But if it is smaller than another negative critical value, the spatial derivative of the corresponding solution blows up in a finite time. Thus it is natural to think about weak solutions in a suitable sense. As a prototype of weak solutions, we consider the Cauchy problem with the Riemann initial data of which the left state is larger than the right state. This condition ensures the existence of the corresponding traveling wave, connecting the left state and the right state asymptotically. This Riemann problem admits a global weak solution, provided that the magnitude of the initial discontinuity is smaller than 1/2. Although the solution has a discontinuity, we have the uniqueness of a solution in weak sense by imposing the entropy condition. Furthermore, the magnitude of the discontinuity contained in the solution decays to zero with an exponential rate as the time t goes to infinity. Also, the solution approaches the corresponding traveling wave with the rate t-1/4 uniformly. The first result is obtained by the maximal principles. To show the second result, we have used an energy method with some estimates, which are also obtained through maximal principles.

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Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i3.470 ◽

2015 ◽

Vol 11 (3) ◽

pp. 3134-3138 ◽

Cited By ~ 3

Author(s):

Mostafa Khater ◽

Mahmoud A.E. Abdelrahman

Keyword(s):

Elliptic Function ◽

Traveling Wave ◽

Evolution Equations ◽

Nonlinear Evolution ◽

Traveling Wave Solutions ◽

Expansion Method ◽

Mathematical Physics ◽

Nonlinear Evolution Equations ◽

Jacobian Elliptic Function ◽

Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity andÂ reliability of the method are tested by its applications to the Couple Boiti-Leon-PempinelliÂ System which plays an important role in mathematical physics.

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Classification of Traveling Wave Solutions of Reaction-Diffusion Systems.

10.21236/ada167101 ◽

1985 ◽

Cited By ~ 5

Author(s):

Konstantin Mischaikow

Keyword(s):

Traveling Wave ◽

Traveling Wave Solutions ◽

Reaction Diffusion ◽

Reaction Diffusion Systems ◽

Wave Solutions ◽

Diffusion Systems

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Traveling wave solutions of the one-dimensional Boussinesq paradigm equation

10.1063/1.4827243 ◽

2013 ◽

Author(s):

V. M. Vassilev ◽

P. A. Djondjorov ◽

M. Ts. Hadzhilazova ◽

I. M. Mladenov

Keyword(s):

Traveling Wave ◽

Traveling Wave Solutions ◽

One Dimensional ◽

Wave Solutions ◽

The One

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Boundary layer analysis for a 2-D Keller-Segel model

Open Mathematics ◽

10.1515/math-2020-0093 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1895-1914

Author(s):

Linlin Meng ◽

Wen-Qing Xu ◽

Shu Wang

Keyword(s):

Boundary Layer ◽

Diffusion Coefficient ◽

Approximate Solution ◽

Chemical Diffusion ◽

Singular Perturbation Theory ◽

Energy Estimates ◽

Chemical Diffusion Coefficient ◽

Boundary Layer Problem ◽

Classical Energy ◽

Layer Analysis

Abstract We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.

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Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

Symmetry ◽

10.3390/sym13020224 ◽

2021 ◽

Vol 13 (2) ◽

pp. 224

Author(s):

Ghaylen Laouini ◽

Amr M. Amin ◽

Mohamed Moustafa

Keyword(s):

Lie Group ◽

Traveling Wave ◽

Invariant Solution ◽

Traveling Wave Solutions ◽

Group Method ◽

Lie Group Method ◽

Reduced Equations ◽

Wave Solutions ◽

Negative Order ◽

Petviashvili Equation

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

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