工程数学学报
工程數學學報
공정수학학보
CHINESE JOURNAL OF ENGINEERING MATHEMATICS
2014年
1期
84-92
,共9页
抛物系统%整体存在性%爆破%爆破率
拋物繫統%整體存在性%爆破%爆破率
포물계통%정체존재성%폭파%폭파솔
parabolic system%global existence%blow-up%blow-up rate
本文研究一类具有齐次Dirichlet边界条件与非局部源项耦合的半线性抛物型方程组的初边值问题．该问题来源于可燃混合气体的热传播过程等自然现象．利用椭圆问题的特征值与特征函数理论和上解下解方法，我们给出了该问题解的临界指标p=p1p2···pk-1．征得当p≤0，且pi >qi，以及系数ri充分小时，问题存在整体解．而如果p>0，则对于充分大的初值，我们证得解在有限时刻爆破．另外，我们证明获得了爆破率的上界．
本文研究一類具有齊次Dirichlet邊界條件與非跼部源項耦閤的半線性拋物型方程組的初邊值問題．該問題來源于可燃混閤氣體的熱傳播過程等自然現象．利用橢圓問題的特徵值與特徵函數理論和上解下解方法，我們給齣瞭該問題解的臨界指標p=p1p2···pk-1．徵得噹p≤0，且pi >qi，以及繫數ri充分小時，問題存在整體解．而如果p>0，則對于充分大的初值，我們證得解在有限時刻爆破．另外，我們證明穫得瞭爆破率的上界．
본문연구일류구유제차Dirichlet변계조건여비국부원항우합적반선성포물형방정조적초변치문제．해문제래원우가연혼합기체적열전파과정등자연현상．이용타원문제적특정치여특정함수이론화상해하해방법，아문급출료해문제해적림계지표p=p1p2···pk-1．정득당p≤0，차pi >qi，이급계수ri충분소시，문제존재정체해．이여과p>0，칙대우충분대적초치，아문증득해재유한시각폭파．령외，아문증명획득료폭파솔적상계．
The initial-boundary value problem of the semilinear parabolic system coupled via nonlocal sources, subjecting to the homogeneous Dirichlet boundary condition, is investigated in this paper. The problem is derived from the natural phenomena such as heat propagation of the combustible mixture gases. The critical exponent p=p1p2 ···pk-1 of this problem is gained by means of the eigenvalue and eigenfunction theory and the methods of supersolution and subsolution. It is proved that under the condition that p ≤ 0, pi > qi, and ri is small enough, every solution to the problem is global, whereas if p > 0 and the initial data are sufficiently large, then the solution blows up in a finite time. In addition, the upper bound of the blow-up rate is also obtained.
