Existence and asymptotic behavior of solutions for quasilinear parabolic systems (Q5891150)

The author considers the following coupled system of quasilinear equations NEWLINE\[NEWLINE\begin{aligned} & \partial u_{i}/\partial t-\nabla \cdot \left( a_{i}D_{i}\left( u_{i}\right) \nabla u_{i}\right) +\mathbf{b}_{i}\cdot \left( D_{i}\left( u_{i}\right) \nabla u_{i}\right) =f_{i}\left( t,x,\mathbf{u}\right) ,t>0,x\in \Omega,\\ & D_{i}\left( u_{i}\right) \partial u_{i}/\partial \nu =g_{i}\left( t,x, \mathbf{u}\right), t>0,x\in \partial \Omega, \\ & u_{i}\left( 0,x\right) =\psi _{i}\left( x\right) ,x\in \Omega,i=1,\dots,N,\end{aligned} NEWLINE\]NEWLINE where \(\mathbf{u\equiv }\left( u_{1},\dots,u_{N}\right),\Omega \) is a bounded domain in \(\mathbb{R}^{n}\) with boundary \(\partial \Omega ,\partial /\partial \nu \) denotes the outward normal derivative on \(\partial \Omega\). It is assumed that the boundary \(\partial \Omega \) is of class \(C^{1+\alpha }\). It is also assumed that, for each \(i=1,\dots,N\), the functions \(a_{i}\equiv a_{i}\left( t,x\right) ,\mathbf{b}_{i}\equiv \mathbf{b}_{i}\left( t,x\right),f_{i}\left( t,x,\cdot \right) ,g_{i}\left( t,x,\cdot \right) \) are Hölder continuous in \(\left[ 0,\infty \right) \times \overline{\Omega }\). The density-dependent diffusion coefficient \(D_{i}\left( u_{i}\right) \) may have the property \(D_{i}\left( 0\right) =0\), which means that the elliptic operators are degenerate. The author's aim is to study the existence, uniqueness and asymptotic behavior of \(\left( 1\right) \) with the mixed quasimonotone reaction functions.NEWLINENEWLINE After a brief introduction, in Section 2, the author shows the existence and uniqueness of \(\left( 1\right) \) by the method of coupled upper and lower solutions, and its associated monotone iterations. In Section 3, the asymptotic behavior of the solutions is investigated. In Section 4, considering the Lotka-Volterra competition model with density-dependent diffusion, the author gives necessary and sufficient conditions such that the positive uniform equilibrium is globally asymptotic stable. Section 5 is devoted to some discussions about the biological and mathematical senses. In this respect, the author shows that the study is a starting attempt to consider the role of density-dependent diffusion on asymptotic behavior of solutions of the Lotka-Volterra model. In fact, the results obtained by the author in Section 4, imply that if the rates of inter-specific competition are large, the long-term behaviors of the species tend to positive uniform equilibrium. The two competitive species are coexistent. The results also have applicability to three species model.