Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential
Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential
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DOI：
10.1080/03605300802402633
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发表时间：
2008-10
影响因子：
1.9
通讯作者：
S. Ervedoza
中科院分区：
文献类型：
--
作者：
S. Ervedoza
本文的目的是分析具有奇异位势− μ/的抛物型方程的控制性质。|X| 2，其中μ是真实的数。当μ ≤（N − 2）2/4时，在[19]中证明了方程可以通过围绕奇点的分布式控制被控制到零。在目前的工作中，使用Carleman估计，我们将证明这个假设是不必要的，我们可以控制方程从任何开子集的热方程。然后，我们将研究μ >（N − 2）2/4的情况，并证明情况完全改变：实际上，我们将考虑正则化势序列μ/（|X| 2 +<$2），并证明了我们不能稳定相应的系统一致关于<$0，由于爆炸模式的存在，集中在奇点。
The goal of this article is to analyze control properties of parabolic equations with a singular potential − μ/|x|2, where μ is a real number. When μ ≤ (N − 2)2/4, it was proved in [19] that the equation can be controlled to zero with a distributed control which surrounds the singularity. In the present work, using Carleman estimates, we will prove that this assumption is not necessary, and that we can control the equation from any open subset as for the heat equation. Then we will study the case μ > (N − 2)2/4, and prove that the situation changes completely: indeed, we will consider a sequence of regularized potentials μ/(|x|2 + ϵ2), and prove that we cannot stabilize the corresponding systems uniformly with respect to ϵ > 0, due to the presence of explosive modes which concentrate around the singularity.