On the Cauchy problem for a reaction–diffusion equation with a singular nonlinearity
On the Cauchy problem for a reaction–diffusion equation with a singular nonlinearity
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DOI：
10.1016/j.jde.2007.06.012
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发表时间：
2007-09
影响因子：
2.4
通讯作者：
Zongming Guo;Juncheng Wei
中科院分区：
文献类型：
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作者：
Zongming Guo;Juncheng Wei
我们考虑以下具有奇异非线性的柯西问题，其中 n⩾3（且 phi 具有正下界）。我们在初始值 phi 上找到了一些条件，使得 (P) 的局部解在有限时间内消失。同时，我们获得了全局存在的最优条件，并研究了这些全局解的大时间行为。特别是，我们证明，如果 ν>0 且 n⩾3，其中 us 是 (P) 且 γ>1 的奇异平衡，则 (P) 具有（唯一）全局经典解 u 且 u⩾γusand 另一方面，研究了 (P) 稳态的正径向解的结构，并获得了正解的一些有趣的性质。此外，还讨论了(P)稳态正径向解的稳定性和弱渐近稳定性。
We consider the following Cauchy problem with a singular nonlinearity with n⩾3 (and ϕ having a positive lower bound). We find some conditions on the initial value ϕ such that the local solutions of (P) vanish in finite time. Meanwhile, we obtain optimal conditions on ϕ for global existence and study the large time behavior of those global solutions. In particular, we prove that if ν>0 and n⩾3, where usis a singular equilibrium of (P) and γ>1, then (P) has a (unique) global classical solution u with u⩾γusand On the other hand, the structure of positive radial solutions of the steady-state of (P) is studied and some interesting properties of the positive solutions are obtained. Moreover, the stability and weakly asymptotic stability of the positive radial solutions of the steady-state of (P) are also discussed.