ABSTRACT

The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential: ${\mathrm{\partial }}_{t}u(x,t)-\mathrm{d}\mathrm{i}\mathrm{v}\left(p\right(x\left)\mathrm{\nabla }u\right(x,t\left)\right)-\left(\mu /|x{|}^2\right)u(x,t)=f(x,t).$ Here μ is a real constant. It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when $0\le \mu \le p_1(n-2{)}^2/4$, and in this paper, we mainly consider the case $0\le \mu <\left(p_1^2/p_2\right)(n-2{)}^2/4$, where $p_1,p_2$ are two positive constants which satisfy: $0<p_1\le p\left(x\right)\le p_2,\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$. We extend the specific Carleman estimates in the paper of Ervedoza [Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun Partial Differ Equ. 2008;33:1996–2019] and Vancostenoble [Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287–1317] to the system. We obtain that we can control the equation from any non-empty open subset as for the heat equation. Moreover, we will study the case $\mu >p_2(n-2{)}^2/4$. We consider a sequence of regularized potentials $\mu /(|x{|}^2+{ϵ}^2),$ and prove that we cannot stabilize the corresponding systems uniformly with respect to $ϵ>0.$.

摘要

本文的目的是分析抛物线​​方程的控制特性，该方程在主部分具有可变系数且具有奇异的平方反比势：${\mathrm{\partial }}_{吨}你(X,吨)-\mathrm{d}\mathrm{一世}\mathrm{v}\left(p\right(X\left)\mathrm{\nabla }你\right(X,吨\left)\right)-\left(\mu /|X{|}^2\right)你(X,吨)=F(X,吨).$这里μ是一个实常数。Goldstein 和 Zhang (2003) 的论文证明了该方程是适定性的，当$0\le \mu \le p_1(n-2{)}^2/4$，在本文中，我们主要考虑这种情况$0\le \mu <\left(p_1^2/p_2\right)(n-2{)}^2/4$， 在哪里$p_1,p_2$是两个满足的正常数：$0<p_1\le p\left(X\right)\le p_2,\mathrm{\forall }X\in \overline{\mathrm{\Omega }}$. 我们在 Ervedoza 的论文中扩展了特定的 Carleman 估计[具有平方反比势的奇异热方程的控制和稳定特性。公用偏微分方程。2008;33:1996–2019] 和 Vancostenoble [奇异抛物方程逆源问题中的 Lipschitz 稳定性。公用偏微分方程。2011;36(8):1287–1317] 到系统。我们得出，我们可以从任何非空的开放子集控制方程，就像热方程一样。此外，我们将研究这个案例$\mu >p_2(n-2{)}^2/4$. 我们考虑一系列正则化势$\mu /(|X{|}^2+{\epsilon }^2),$并证明我们不能一致地稳定相应的系统$\epsilon >\mathrm{0。}$.