渗流方程Cauchy问题的非全局解

A Local Solution for the Cauchy Problem of a Porous Medium Equation

对稀疏介质中的渗流方程:(|u(x,t)|^(n-1)u)_i=△u,0<m<1,当初值u_0(x)无界且具一定增长阶时,本文证明存在古典解u(x,t)∈C^(2.1)(Qre),当|x|→∞时解具有和初值相同的增长速度,当t→T^-_0时,解发生Blow up。

For the Cauchy problem of the porous medium equation (|u(x,t)|m-2u)t = △u with initial data which belongs to a certain growth class, the existence of local (in time) classial solution u (x,t)∈C1,2(Qr0) is obtained. The solution may grow as |x|→∞ at the same growth rate as the initial data's and may blow up as t→T0- for every x∈RN\{0}.