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Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems. (English) Zbl 1192.35093

Summary: This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: \(u_{1t}= f_1(u_2)(\Delta u_1+a_1u_1),\dots, u_{(n-1)t}= f_{n-1}(u_n)(\Delta u_{n-1}+a_{n-1} u_{n-1})\), \(u_{nt}=f_n(u_1)(\Delta u_n+a_nu_n)\) with homogeneous Dirichlet boundary condition and positive initial condition, where \(a_i\) \((i=1,2,\dots,n)\) are positive constants and \(f_i\) \((i=1,2,\dots,n)\) satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it is proved that: (i) when \(\min\{a_1,\dots,a_n\}\leq\lambda_1\) 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 \(\min\{a_1,\dots, a_n\}>\lambda_1\), and the initial datum \((u_{10},\dots,u_{n0})\) 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.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35K65 Degenerate parabolic equations
35B44 Blow-up in context of PDEs
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