A degenerate and strongly coupled quasilinear parabolic system not in divergence form

This paper deals with positive solutions of degenerate and strongly coupled quasilinear parabolic system not in divergence form: u(t) = v(P)(Deltau + au), v(t) = u(q)(Deltau + bv) with null Dirichlet boundary condition and positive initial condition, where p, q, a and b are all positive constants, and p, q greater than or equal to 1. The local existence of positive classical solution is proved. Moreover, it will be proved that: (i) When min{a, b} less than or equal to 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 (we can not prove the uniqueness result in general); (ii) When min{a, b} > lambda(1), there is no global positive classical solution (we can not still prove the uniqueness result), if in addition the initial datum (u(0), v(0)) satisfies Deltau(0) + au(0) greater than or equal to 0, Deltav(0) + bv(0) greater than or equal to 0 in Omega, 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.