Convergence rate estimates of a higher-dimension reaction-diffusion system with density-dependent motility (Q2155964)

This paper considers a three-component parabolic system with density-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain: \[ \begin{cases} u_t=\Delta(\gamma(v)u)+\alpha uF(w)-\theta u^{r}, & x \in \Omega, \, t>0,\\ v_t=D \Delta v-v+u, & x \in \Omega, \, t>0,\\ w_t= \Delta w-u F(w), & x \in \Omega, \, t>0,\\ \frac{\partial u}{\partial \nu}= \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu}=0, & x \in \partial \Omega, \, t>0, \\ (u,v,w)(x,0)=(u_0, v_0, w_0)(x), & x \in \Omega, \end{cases} \] where \(\alpha,\, \theta\geq 0\), and \begin{itemize} \item \(\gamma(v)\in C^3([0,\infty))\), there exist \(\gamma_1, \gamma_2, \eta>0\) such that \(0<\gamma_1\leq \gamma(v)\leq\gamma_2\) and \(|\gamma'(v)|\leq \eta\) for all \(v\geq0\). \item \(F(w)\in C^1([0,\infty))\), \(F(0)=0\), \(F(w)>0\) in \((0,\infty)\) and \(F'(w)>0\) on \([0, \infty)\). \end{itemize} The main contribution of this paper is to establish the global boundedness, large-time behavior and exponential convergence rate of the classical solutions to higher-dimension \((n\geq3)\) reaction-diffusion system under suitable conditions. Meanwhile, the author proved the decay rate solution to the system with \(r=1\) in two dimensions. The proof of the theorem is delicate and seem to be right. This paper improves the result of \textit{H.-Y. Jin} et al. [J. Differ. Equations 269, No. 9, 6758--6793 (2020; Zbl 1441.35142)] on the decay rate of solutions to the system with \(r=1\) in two dimensions. This is a great work. The paper is well organized with a complete list of relevant references. The presentation of the paper is very clear. However, the condition of \(\gamma(v)\) and \(F(w)\) may be well improved.