Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

This paper is concerned with the existence and nonlinear stability of traveling wave solutions to a chemotaxis model with logarithmic sensitivity and nonlinear production \begin{equation}\tag{$\star$}\left\{\begin{aligned}& u_t- (uv)_x=D\, u_{xx}, \\& v_t+ \left( \varepsilon v^2-u^2 \right)_x=\varepsilon\, v_{xx},\end{aligned}\right.\end{equation}for $x\in \mathbb{R}$, $t\geqslant 0$, $D>0$, $\varepsilon>0$, and the initial data\begin{equation*}(u,v)(x,0)=(u_0,v_0)(x)\longrightarrow\left\{\begin{aligned}&(u_{-},v_{-}), & x\to -\infty , \\&(u_{+},v_{+}), & x\to +\infty.\end{aligned}\right.\end{equation*}Based on the phase plane analysis method, we establish the existence of traveling fronts to the system ($\star$). The asymptotic nonlinear stability of traveling wave solutions to the system ($\star$) is identified for $u_{+}>0$ and $u_{+}=0$ by weighted energy estimate methods, respectively. Compared with the previous results, our results do not require any smallness assumption on the wave strengths.

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On a chemotaxis model with singular sensitivity: Convergence rate towards spiky steady state

Feng,

Tang,

Wang

et al. 2024

DCDS

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Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

Cited by 3 publications

References 35 publications

Global stability of a system of viscous balance laws arising from chemotaxis with dynamic boundary flux

Global stability of a system of viscous balance laws arising from chemotaxis with dynamic boundary flux

Asymptotic nonlinear stability of traveling waves to a chemotaxis model with logarithmic sensitivity and nonlinear production

On a chemotaxis model with singular sensitivity: Convergence rate towards spiky steady state

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