具非局部源的非线性高阶抛物型方程解的全局存在性与非全局存在性

Global and Non-global Existence of Solutions for a Higher-order Nonlinear Parabolic Equation with Nonlocal Source

本文考虑一类具非局部源的高阶抛物型方程u_(t)=-(-Δ)^(m)u+(∫_(R^(n))|u|^(1+σ)dy)^((p-1)/(1+σ))|u|^(r)的Cauchy问题.近年来,我们已给出这一方程的Fujita临界指标p_(c)=1+((2m-n(r-1))(1+σ))/(nσ),即当1<p≤p_(c)时,对任意初值,解都在有限时刻发生爆破;当p> p_(c)时,存在非全局解也存在全局解,这取决于初值的大小.本文进一步确定了这一方程的第二临界指标a^(*)=(2m+(n(p-1))/(1+σ))/(p+r-2),用于在p>p_(c)这一全局解与非全局解的共存区域内鉴别初值的大小.我们发现:(1)与具局部源|u|^(p)的高阶抛物型方程的第二临界指标a^(*)=2/(p-m)不同的是,这里与空间维数n有关;(2)非局部源中参数σ的增大有利于扩大解的全局存在区域.

This paper studies the higher-order nonlocal parabolic equation u_(t)=-(-Δ)^(m)u+(∫_(R^(n))|u|^(1+σ)dy)^((p-1)/(1+σ))|u|^(r).Recently we obtained the critical exponent p_(c)=1+((2m-n(r-1))(1+σ))/(nσ)to the problem,i.e.,any solution blows up in finite time whenever 1 c,and there are both global and non-global solutions if p>pc.This paper will determine the second critical exponent to identify the global and non-global solutions in the co-existence region p>pc that a^(*)=(2m+(n(p-1))/(1+σ))/(p+r-2),quite different from that for the parallel problem with local source|u|^(p),where a^(*)=2/(p-m)is independent of the dimension n.In addition,the second critical exponent of this paper shows that the global existence region of solutions would be enlarged when the nonlocal parameterσis increasing.