Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: $u_{1t}=f_1(u_2)(\Delta u_1+a_1u_1),\cdots, u_{(n-1)t}=f_{n-1}(u_n)(\Delta u_{n-1}+a_{n-1} u_{n-1}),\ u_{nt}=f_n(u_1)(\Delta u_n+a_nu_n)$ with homogeneous Dirichlet boundary condition and positive initial condition, where $a_i\ (i=1,2,\cdots,n)$ are positive constants and $f_i\ (i=1,2,\cdots,n)$ satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when $\min\{a_1,\cdots,\ a_n\}\leq\lambda_1$ then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when $\min\{a_1,\cdots,\ a_n\}>\lambda_1$, and the initial datum $(u_{10},\cdots,\ u_{n0})$ satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where $\lambda_1$ is the first eigenvalue of $-\Delta$ in $\Omega$ with homogeneous Dirichlet boundary condition.