On the analysis of blowup phenomena for a nonlinear parabolic equation

本文研究了一类具有幂非线性的半线性扩散方程的爆破问题。在非凸域上，考虑了方程在Neumann边界条件和Dirichlet边界条件下的爆破率。由Giga Kohn得出的一个著名的结果不能应用于这些情况，所以我们用Liouville类型定理证明了放大是I型的。在这里，如果一个方程的解的爆破率与相应的普通方程的解的爆破率相同，则该方程的解被称为I型爆破。其次，在大扩散情况下，用域的性质描述柯西-诺伊曼问题爆破解集的位置。我们还研究了Sobolev意义上的超临界指数爆破后的延拓。在有限时间内爆炸的解，在爆炸后成为所有时间的经典解，随着时间趋于无穷，具有各种行为。并且，得到了在有限时间内爆破，爆破后立即变为规则，再爆破的解。

In this research, we investigated a blowup problem for a semilinear diffusion equation with power nonlinerity. The blowup rate for the equation under the Neumann boundary condition or the Dirichlet boundary condition in a non-convex domain is considered. A well-lnown result due to Giga Kohn cannot be applied to these cases, so we showed the blowup is of type I making use of Liouville type theorem. Here a solution of the equation is said to exhibit the type I blowup if the blowup rate is as same as that of solutions to the corresponding ordinary equation. Next, the 1 location of the blowup set of solutions to the Cauchy-Neumann problem is described by the property of the domain in the case of large diffusion. We also studied the continuation after blowup for supercritical exponent in the Sobolev sense. Solutions blowing up in finite time and being a classical solution for all time after blowup with various behaviors as time tends to infinity. Moreover, we got a solution which blows up in finite time and becomes regular immediately after the blowup time and blows up again.