带局部非线性反应项的退化抛物方程解的爆破性质

The Blowup Property of Solutions to Degenerate Parabolic Equation with Localized Nonlinear Reactions

本文研究带局部非线性反应项的退化抛物方程解的爆破性质ut=△um+up(x0,t)-kuq(x,t),其中p≥q>0,p>1,0<m<p(0<k<1,如果p=q>1),x0是有界区域Ω内的固定点,Ω(?)RN.在一定的假设条件下,证明了解在有限时刻爆破并且爆破点集是整个区域Ω.另外,如果解u(·,t)是径向对称函数且ur≤0,则解在接近爆破时刻的爆破速率在区域Ω上是一致的.在解是非径向对称的情况下,我们用其他技巧证得解的整体爆破性.

In this paper, we investigate the blowup property of solution to degenerate parabolic equations with localized nonlinear reactions where p ≥ q > 0, p > 1, 0<m<p(0<k<1, if p = q > 1) and x0 is a fixed point in the domain Ω(?) RN. We show that under certain conditions the solution blows up in finite time. Moreover, we prove that the set of all blowup points is the whole region. Furthermore, the growth rate of solution near the blowup time is uniform in the domain, provided that u(.,t) are radial functions and ur ≤ 0. We use other techniques to prove the global blowup in the non-symmetric case.