Boundedness and asymptotic behavior in a quasilinear chemotaxis system with nonlinear diffusion and singular sensitivity for alopecia areata

Abstract

This paper is concerned with a three-component quasilinear chemotaxis system with nonlinear diffusion and singular sensitivity for alopecia areata $\left(\begin{array}{cc}{u}_t=\nabla \left(D_1\left(u\right)\nabla u\right)-{\chi }_1\nabla \left(\frac{u}{w}\nabla w\right)+w-{\mu }_1{u}^{{\gamma }_1},& (x,t)\in \Omega \times (0,\infty ),\\ v_t=\nabla \left(D_2\left(v\right)\nabla v\right)-{\chi }_2\nabla \left(\frac{v}{w}\nabla w\right)+w+ruv-{\mu }_2{v}^{{\gamma }_2},& (x,t)\in \Omega \times (0,\infty ),\\ w_t=\Delta w+u+v-w,& (x,t)\in \Omega \times (0,\infty ),\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0,& (x,t)\in \partial \Omega \times (0,\infty ),\\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),w(x,0)=w_0\left(x\right),& x\in \Omega ,\end{array}\right)$associated with homogeneous Neumann boundary conditions in a convex smooth bounded domain $\Omega \subset R^2$. For $i=1,2,$ the parameters ${\chi }_i,{\mu }_i,r$ are positive and ${\gamma }_i\ge 2$. The nonlinear diffusion functions $D_i\left(s\right)\in C^2$ satisfy $D_i\left(s\right)⩾{(s+1)}^{{\alpha }_i}$ for all $s\ge 0$. We delve into analyzing the global existence and boundedness of classical solutions for the aforementioned system under specific conditions. Additionally, in the scenario where ${\gamma }_i=2$, we develop a Lyapunov functional and scrutinize its temporal evolution to ascertain the asymptotic stability of the coexistence state.