一类半线性抛物方程组正解的整体存在性与非存在性

Existence and Nonexistence of Global Positive Solutions to a Class Semilinear Parabolic Systems

考察了半线性抛物方程组:(ui)t=Δui+|x|miuipi+1,(x,t)∈RN×(0,T),ui(x,0)=ui0(x),x∈RN,i=1,2,…,s.得到了该方程组的爆破临界指标为1+N2(1+β).当1<γ<1+2/N(1+β),方程组的所有正解都是爆破的;当γ>1+N2(1+β),则在初值ui0(x)较小时方程组存在整体解,而在初值u0i(x)较大时,方程组的任何正解都在有限时间内爆破,这里γ,β由本文(5)式给出.

The following semilinear parabolic systems （ui）t=Δui＋｜x｜miupii＋1,（x,t）∈RN×（0,T）,ui（x,0）=u0i（x）,x∈RN,i=1,2,…,s.are considered.The blowing-up critical exponent of the problem is shown to be 1＋2N（1＋β）.If 1γ1＋2/N（1＋β）,all nonnegative positive solutions will blow up in infinite time;while if γ 1＋2N（1＋β）,there will exist global solution if the initial data u0i（x） are small enough and all nonnegative positive solutions will blow up in infinite time if the initial data u0i（x） are large enough,here,γ,β is defined by（5） as in context.