具奇异系数的反应扩散方程组Cauchy问题

On the Cauchy Problem for a Reaction-Diffusion System with Singular Coefficients

该文讨论如下具有奇异系数的反应扩散方程组Cauchy问题非负局部解的存在性和不存在性,以及解在有限时间内的爆破问题ut- t- 1Δu =α1uq1+β1vp1+ f1( x) , t>0 ,x∈RN;vt- t- 1Δv =α2 uq2 +β2 vp2 + f2 ( x) , t>0 ,x∈RN;limt→0 +u( t,x) =limt→0 +v( t,x) =0 , x∈RN.其中pi>1 ,qi>1 ( i=1 ,2 ) ,α1≥0 ,α2 >0 ,β1>0 ,β2 ≥0 ,fi( x) ( i=1 ,2 )为连续非负有界函数,( f1( x) ,f2 ( x) ) ( 0 ,0 ) .文章给出了非负局部解存在的显式条件和非负局部解不存在的比较结果,也得到解在有限时间爆破的一些结果.

This paper is concerned with the local existence and nonexistence of nonnegative solutions and blow up problem in a finite time for the reaction-diffusion system with singular coefficients u\-t-t\+\{-1\}Δ u=α\-1u\+\{q\-1\}+β\-1v\+\{p\-1\}+f\-1(x), t>0,x∈R\+N, v\-t-t\+\{-1\}Δ v=α\-2u\+\{q\-2\}+β\-2v\+\{p\-2\}+f\-2(x), t>0,x∈R\+N, \{lim\}t→0\++u(t,x)=\{lim\}t→0\++v(t,x)=0, x∈R\+N. where p\-i>1,q\-i>1(i=1,2),α\-1≥0,α\-2>0, β\-1>0,β\-2≥0, and f\-i(x)(i=1,2) are continuous, nonnegative and bounded functions, (f\-1(x),f\-2(x))(0,0). The authors give an explicit condition for the local existence of nonnegative solutions and a comparison result for the local nonexistence of nonnegative solutions of the system. Some blow up results for the system are also obtained.