A 1-d quasilinear nonuniform parabolic chemotaxis model with volume-filling effect (Q2852332)

In the one dimensinal case the following coupled system of quasilinear parabolic equations NEWLINE\[NEWLINEu_t=\nabla\cdot \left\{D(u)\nabla u-B(u)\nabla \upsilon \right\},\,\,\, (x,t) \in \Omega\times(0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\upsilon_{t}=\alpha\Delta\upsilon-\beta\upsilon+\gamma u,\,\,\,\, (x,t)\in \Omega\times(0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINE \nabla u \cdot \overrightarrow{n}=0,\,\,\,\,\, \nabla\upsilon\cdot\overrightarrow{n}=0,\,\,\,\, (x,t)\in \Gamma\times(0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0)=u_{0}(x)\geq 0,\,\,\, \upsilon(x,0)=\upsilon_{0}(x)\geq 0,\,\,\,\, x\in \Omega, NEWLINE\]NEWLINE with NEWLINE\[NEWLINEB(u)=\chi_{0} uq(u)=\chi_{0}u(1+u)^{-\lambda}, \,\,\,\,\,\,\;\lambda>0,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE D(u)=D_{0}(q(u)-uq'(u))=D_{0}\frac{1+(\lambda+1)u}{((1+u)^{\lambda+1}}, NEWLINE\]NEWLINE where \(\Omega \in \mathbb{R}^{n}\) is a bounded domain with smooth boundary \(\Gamma,\alpha,\beta,\gamma,\lambda,\chi_{0},D_{0}\) are given positive constants, and \(\overrightarrow{n}\) denotes the outer normal vector, is studied. The global existence and uniqueness of the classical solution, convergence to equilibrium of the global solution, as time goes to infinity, as well as the convergence rate are proved.