On the singularity of solutions for nonlinear parabolic equation

一类具有幂非线性项的半线性热方程的解在Sobolev临界指数前后的行为发生了显著的变化。它们在亚临界情况下进行了详细的研究，但在超临界情况下只有少数结果。不完全爆破、整体解的增长和II型爆破是超临界情形下的典型现象，不完全爆破解的存在性是已知的，但爆破后的行为是未知的。得到了一个在经典意义下爆破两次的弱解，并将其推广到一般的多重爆破。这些解在每个爆破时刻在同一点（原点）爆破，并且两个连续爆破时刻之间的差异足够大。我们研究了一个解的存在性，对于给定的爆破时刻，该解在不同的爆破时刻在不同的位置爆破，Galaktionov-Vazquez和我证明了存在一个峰值解，该峰值解随着时间的推移收敛到0。利用无穷维动力系统的方法构造了具有精确增长率的增长解。Herrero-Velazquez证明了存在一个解会发生II型爆破。然而，他们的证明是非常漫长和复杂的。我们得到了一个II型爆破解作为整体解和爆破解之间的分界线。我们的证明比前一个简单。

The behavior of solutions to a semilinear heat equation with power nonlinearity dramatically changes before and after the Sobolev critical exponent. They are investigated in detail in the subcritical case, but there are only a few results in the supercritical case. Incomplete blowup, growup of a global solution and type II blowup are typical phenomena in the superciritical case.The existence of a incomplete blowup solution is known, but the behavior after blowup has reminded open. We obtained a weak solution which blowup twice in the classical sense, and extended it to general multiple blowup. These solutions blow up at the same point (at the origin) at each blowup time and the difference between two successive blowup times is sufficiently large. We studied the existence of a solution which blows up at different places at different blowup times for given blowup times.By Galaktionov-Vazquez and myself, there were peaking solution which converges to 0 as time tens to infinity. We gave peaking solutions with various behaviors at time infinity.We constructed growup solutions with exact growup rate applying a method based on infinite-dimensional dynamical system. It also gave the optimal growup rate of solutions below the singular steady state.Herrero-Velazquez showed that there exists a solution which undergoes type II blowup. However their proof is very long and complicated. We got a type II blowup solution as a separatirix between global solutions and blowup solutions. Our proof is simpler than the previous one.