非线性反应扩散方程解的Blow-up问题

BLOW UP BEHAVIOR OF SOLUTIONS OF NONLINEAR REACTION DIFFUSION EQUATIONS

考虑反应扩散方程初边值问题ｕｔ＝Ｌｕ＋ｆ（ｕ），Ω×（ｏ，Ｔ）βｕν＋ｕΩ＝ｈ（ｕ），（Ｅ）ｕ（ｘ，０）＝ｕ０（ｘ）解的Ｂｌｏｗ－ｕｐ问题，其中：Ｌ≡∑ｎｉ，ｊ＝１ｘｉｉｊｘｊ＋∑ｎｉ＝１ｉｘｉ是椭圆算子，β是常数，０＜β＜＋∞，Ω是Ｒｎ中的有界区域，ｕν是关于（ａｉｊ）在Ω上的余法向导数。在ｆ和ｇ的适当条件下，证明了问题（Ｅ）的光滑解只能在一个有界区间〔０，Ｔ０）存在，即有：ｌｉｍｔ→Ｔ－０ｓｕｐｘ∈Ωｕ（ｘ，ｔ）＝＋∞推广了前人的工作。

The Blow up behavior of initial boundary value problem of reaction diffusion equation is considered: ut=Lu+f(u),Ω×(0,T) βuν+u Ω =h(u) u(x,0)=u 0(x)(E) where L≡∑ni,j=1x ia ij x j+∑ni=1a ix i denotes an elliptic openraior, β is a constant and 0<β<+∞,ΩR n assumes bounded and u/ν is a conormal derivative with respect to (a ij ) on Ω .In this paper the blow up behavior of the problem( E ) is studied.Under some conditions on f and g ,the smooth solutions of the broblem( E ) cannot exist on 0,T 0) .Namely,there exists a T 0,0<T 0<+∞ ,such that lim t→T 0 sup x∈Ω u(x,t)=+∞ .