拟线性退化抛物型方程组解的整体存在性和爆破

Global Existence and Blow-up of Solutions to a Quasi-linear Degenerate Parabolic System

该文研究光滑有界区域ΩR^N(N≥1)上具有齐次Dirichlet边界条件的拟线性退化抛物型方程组u_t-div(|▽u|^(p-2)▽u)=av~α,u_t-div(|▽v|^(q-2)▽v)=bu~β的非负解的性质,其中p,q>2,α,β≥1,a,b>0是常数.该文指出上述方程组的解是否在有限时刻爆破依赖于初值、系数a与b以及αβ和(p-1)(q-1)之间的关系.

This paper investigates the nonnegative solutions of quasi-linear degenerate parabolic systemut-div（｜△u｜^p-2△u）=av^α,vt-div（｜△v｜^q-2△v）=bu^β with zero Dirichlet boundary conditions in a smooth bounded domain Ω∩→ R^N（N ≥ 1）, where p, q 〉 2, α,β ≥ 1, a, b 〉 0 are constants. It is obtained that whether the solution blows up in finite time or not depends on the initial data, the coefficients a and b, and the relation between αβ and （p- 1）（q - 1）.