高阶半线性抛物型方程解的整体存在性

Existence of Global Solutions of Higher-Order Semilinear Parabolic Equations

研究了如下高阶半线性抛物型方程的Cauchy问题{ut+(-Δ)mu=│u│p-1u,(x,t)∈Rn×R1+ u(x,0)=u0(x),x∈Rn的解的整体存在性,其中m是正整数,p>1+2m/n,n≥2。首先将该问题转化为与之等价的积分方程,然后通过引入该问题的一个自相似核构造了一个积分方程,该积分方程的解控制了原问题的等价积分方程的解,最后通过证明构造的积分方程的解有界,从而得到等价积分方程的解有界,因此,当m≥2且初值u0(x)满足u0(x)≤α/(1+x2m/(p-1))时,该问题有整体强解。另外在条件lim|x|→∞ inf│x│2m/(p-1)u0(x)>0下,利用弱解的定义和试验函数的紧支性证明了该问题的弱解的负部相对于正部是不能忽略的。

The existence of global solution to the following Cauchy problem for the higher-order semilinear parabolic equation is studied in this paper： {u（x,0）=u0（x）,x∈R^n ut＋（-△）^mu=｜u｜^p-1u,（x,t）∈R^n×R^1 where p 〉 1 ＋ 2m/n and m is a positive integer. First, the problem is transformed into an equivalent integral equation, then another integral equation, whose solution can control the solution of the equivalent integral equation, is constructed by introducing a self-similar kenel function. Finally, the boundedness of the equivalent integral equation can be obtained by proving the boundedness of the integral equation structured. Thus, if m≥2 and uo （x） satisfies ｜u0（x）｜≤a/（1＋｜x｜^2m/（p-1）, the solution of the problem is global. Besides, if ｜x｜→∞lim inf｜x｜^2m/（p-1）u0（x）〉0 holds, then using the definition of the weak solution and the compactness of test function, the negative part of the weak solution can not be ignored with respect to the positive part.