【作者】 李锋杰；

【导师】 郑斯宁；

【作者基本信息】 大连理工大学 ， 计算数学， 2006， 博士

【摘要】 本文主要研究一类经由非线性内部源和边界流多重耦合的抛物型方程组（问题（Ⅰ）,问题（Ⅱ））,详细讨论了其非整体解的同时与不同时blow-up的条件、同时blow-up速率和blow-up集等问题.由于非线性的多重性,对所考虑的模型分别得到多达四种（对应问题（Ⅰ））或九种（对应问题（Ⅱ））不同的同时blow-up速率,问题相当复杂。这里通过引入相应的特征代数方程组,给出了blow-up临界指标和同时blow-up速率的简捷刻画,得到完整的结果。特别地,在同时与不同时blow-up的讨论中首次发现在同一指标条件下,不同的初值可以导致不同的同时blow-up速率的有趣现象。此外,还讨论了具有指数型内部吸收项和耦合边界流的非线性抛物型方程组（问题（Ⅲ））。通过所引入的特征代数方程组的解的倒数的符号,清晰刻画出区分整体解与非整体解的临界指标。对问题（Ⅲ）的相应单个方程情形,还进一步得到了非整体解的blow-up profile。 问题（Ⅰ）其中参数σ₁,σ₂∈{0,1},区域Ω（?）R^N,且边界光滑,u₀,υ₀满足相容性条件的正光滑函数。对应于σ₁,σ_∈{0,1},问题（Ⅰ）包含三种典型情形: 当σ₁=σ₂=1时,问题（Ⅰ）变为Souplet和Tayachi,Rossi和Souplet分别研究了相应于该情形的Cauchy问题以及齐次Dirichlet问题,讨论了同时与不同时blow-up的条件,以及同时与不同时blow-up共存的指标区域,并得到两种同时blow-up速率。 本文主要讨论问题（Ⅰ）的其它两种典型情形,采用与不同的方法,得到了更多有趣的结果。更多还原

【Abstract】 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 ?R^N 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 = <r2 = 0, Problem （I） turns into the heat equations coupled in nonlinear boundary fluxIf o\ = 1, 02 = 0 （or G\ — 0, 02 = 1, similarly）, Problem （I） becomes across-coupled system of the formut = Aw + uu（x, 0） =mvt = Av, {x,t)€Qx（0,T）,= ug + vnj （x,t） e dQ x （0,T）, v（x, 0） = vQ（x）, x e fi.We establish interesting results for the above two models, such as the necessary and sufficient conditions for simultaneous blow-up only, the conditions for non-simultaneous blow-up only, the parameter regions for coexistence of simultaneous and non-simultaneous blow-up, the four possible simultaneous blow-up rates, and the blow-up sets. In particular, it has been the first time to find that, under the same requirement on exponent parameters, different initial data may lead to different simultaneous blow-up rates.Next, we consider the more complicated case with multi-coupled nonlinearities （Problem （II）） for iV = 1, and obtain similar results as those in Problem （I）.Problem （II）The third problem discussed is a coupled nonlinear parabolic system with nonlinear inner absorptions and nonlinear boundary flux: Problem （III）It will be proved that the critical exponent is determined by the signs of reciprocals of solutions to the associated characteristic algebraic system. Furthermore, the exact blowup profile is established for the scalar case of Problem （III）.更多还原