带非局部源的退化奇异半线性抛物方程的爆破

Blow-up for Degenerate and Singular Semilinear Parabolic Equations with Nonlocal Source

本文研究带齐次Dirichlet边界条件的非局部退化奇异半线性抛物方程ut-(xαux)x=∫0af(u)dx在(0,a)×(0,T)内正解的爆破性质,建立了古典解的局部存在性与唯一性.在适当的假设条件下,得到了正解的整体存在性与有限时刻爆破的结论.本文还证明了爆破点集是整个区域,这与局部源情形不同.进而,对于特殊情形:f(u)=up,p>1及,f(u)=eu,精确地确定了爆破的速率.

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate and singular semilinear parabolic equation ut - (xαux)x = ∫0af(u)dx in (0, a)×(0, T) with homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. It is also proved that the blow-up set is the whole domain, which differs from the local case. Furthermore, the blow-up rate is precisely determined for that in the special cases: f(u) = up, p > 1 and f(u) = eu.