| This thesis mainly deals with two nonlinear parabolic systems, both multi-coupled via nonlinear inner sources and nonlinear boundary flux (Problem (I), (II)). The simultaneous and non-simultaneous blow-up, blow-up rates and sets are studied in detail. The multi-coupled nonlinearities there result in four (for Problems (I)) or nine (for Problems (II)) different simultaneous blow-up rates, which are briefly described by the characteristic algebraic systems associated to the two problems respectively. It is interesting to find that different initial data may lead to different simultaneous blow-up rates even in the same region of the exponent parameters. The third problem (Problem (III)) studied in this thesis is a nonlinear parabolic system multi-coupled via nonlinear exponent-type inner absorptions and boundary flux. The critical exponent is determined by the signs of reciprocals of solutions to the associated characteristic algebraic system. In addition, the exact blow-up profile is established for the scalar case of Problem (III).Problem (I)where parameters σ1,σ2 ∈ {0,1}, Ω ?RN is a bounded domain with smooth boundary, u0, u0 are positive and smooth functions satisfying the compatibility conditions. Problem (I) covers three typical types:The Cauchy problem and the homogeneous Dirichlet problem corresponding to this case were studied by Souplet and Tayachi''4', Rossi and Souplet'""', respectively, for which the conditions of simultaneous and non-simultaneous blow-up, the regions of coexistence, and two kinds of simultaneous blow-up rates were obtained.In this thesis, a detailed study will be given to the other two cases of Problem (I). The methods applied here are different from those in [74, 69], while the results obtained seem even more rich and interesting.If crj = |
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