Study on the asymptotic behavior of solutions of quasilinear parabolic equations with a blow-up term

在我们的项目中，我们研究了Dirichlet问题非负解的渐近性态（Ω是有界的）或柯西问题（Ω = R^N）的拟线性抛物方程与热源：u_t-Δu^m= F in（x，t）∈ Ω ×（0，T），其中m ≠ 1，且F =f（u）（通常的热源）或F = f（u（x_0（t），t））（x_0（t）∈Ω）（定域反应）。这个方程代表了各种现象，并给出了各种有趣的问题。我们已经得到了这些问题的下三个结果。(i)当m=1，Ω是有界区域，F=f（u（x_0（t），t））时，我们证明了整体解的有界性是由x_0（t）在t→∞时的渐近行为决定的，这个结果是我们关于所有解的分类结果的一部分.然而，当m> 1时，我们对这个问题没有好的结果，因为我们不知道解的唯一性是否成立。（ii）当m <$1，Ω= R^N，F=u^P时，我们研究了解在无穷远处爆破的精确性态。特别地，我们引入了“具有最小爆破时间的爆破解”，并证明了这样的解在空间无穷远处爆破。我们给出了一个解是爆破解且爆破时间最短的充要条件。我们还给出了一个爆破解以最少爆破时间在方向上爆破的充要条件。（iii）当m>1，Ω= R^N，F = u^P时，研究了解在有限时间内爆破的条件，得到了新的结果。

In our project, we study the asymptotic behavior of nonnegative solutions of the Dirichlet problem(Ω is bounded) or the Cauchy problem(Ω = R^N) for a quasilinear parabolic equation with a heat source : u_t-Δu^m= F in(x, t)∈ Ω ×(0, T), where m ≧1, and F =f(u)(a usual heat source) or F = f(u(x_0(t), t))(x_0(t)∈Ω)(localized reaction).Furthermore, we assume that f satisfies some blow-up condition. This equation represents various phenomena and gives interesting various problems. We have obtained the next three results for these problems.(i) When m=1, Ω is a bounded domain and F=f(u(x_0(t), t)), we showed that the boundedness of global solutions is determined by the asymptotic behavior of x_0(t)as t→∞.This result is a part of our result on the classification of all solutions. However, when m> 1, we do not have good results for this problem, since we do not know whether or not the uniqueness of solutions holds.(ii)When m ≧1, Ω= R^N and F=u^P , we studied the precise behavior of solutions which blow up at space infinity. In particular, we introduced "blow-up solution with the least blow-up time" and showed that such a solution blows up at space infinity. We give a necessary and sufficient condition for a solution to be a blow-up solution with the least blow-up time. We also give a necessary and sufficient condition for a blow-up solution with the least blow-up time to blow up in a direction ψ.(iii)When m>1, Ω= R^N and F = u^P , we studied under what condition the solution blows up in finite time, and got new results.