The null controllability for a singular heat equation with variable coefficients

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 $ϵ>0.$.

Keywords:

• Singular heat equation
• null controllability
• Carleman estimate
• observability

2010 Mathematics Subject Classifications:

• 35B37
• 93B05
• 35K05
• 49J20

Acknowledgements

The first author is supported by the scholarship from China Scholarship Council (CSC).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by the scholarship from China Scholarship Council (CSC) [201706340180].