具非线性边值条件的热方程组解的全局存在性与爆破

Global Existence and Blow-up for a System of Heat Equations with Nonlinear Boundary Conditions

证明 Rn 中有界区域Ω上具非线性边值条件的热方程组 ut=Δu,vt=Δv,x∈Ω ,0 <t<T, u n=h1( t) up11vp12 , v n=h2 ( t) up2 1vp2 2 ,x∈ Ω ,0 <t<T, u( x ,0 ) =u0 ( x) >0 ,v( x,0 ) =v0 ( x) >0 ,x∈Ω正解全局存在的充要条件为 0 <α≤ 1,这儿 hi( t) =( t+t0 ) σi 或 eσit( i =1,2 ) ,t0 >0 ,pij ≥ 0 ,p12 . p2 1≠ 0 ,σi 为任意实数 ,α =p11+bp12 ,b =( 1+p2 1- p11) / ( 1+p12 - p2 2 ) >0 .

It is proved that the necessary and sufficient condition is 0<α≤1 for the global existence of positive solutions to the following heat equations with nonlinear boundary conditions in a bounded domain Ω in R n  u t=Δu, v t=Δv, x∈Ω, 0<t<T un=h 1(t)u p 11 v p 12 , vn=h 2(t)u p 21 v p 22 , x∈Ω, 0<t<T, u(x,0)=u 0(x)>0, v(x,0)=v 0(x)>0, x∈Ω where h i(t)=(t+t 0) σ i  or e σ it (i=1,2), t 0>0, p ij ≥0, p 12 ·p 21 ≠0, σ i(i=1,2) are real numbers, α=p 11 +bp 12 , b=(1+p 21 -p 11 )/(1+p 12 -p 22 )>0.