Quantitative Weak Unique Continuation on Annular Domains for Backward Degenerate Parabolic Equations with Degenerate Interior Points

Technical Summary: Quantitative Weak Unique Continuation on Annular Domains for Backward Degenerate Parabolic Equations with Degenerate Interior Points

Problem Statement
This paper investigates the weak unique continuation property for backward degenerate parabolic equations featuring a degenerate interior point. The specific equation under study is:
$$\begin{cases} \partial_t \phi + \text{div}(|x|^\alpha \nabla \phi) = 0, & \text{in } Q = \Omega \times (0, T), \\ \phi = 0, & \text{on } \partial Q, \\ \phi(T) = \phi_T, & \text{in } \Omega, \end{cases}$$
where $\alpha \in (0, 2)$, $\Omega \subset \mathbb{R}^N$ ($N \ge 2$) is a bounded domain containing the origin with a $C^2$ boundary, and $\phi_T \in L^2(\Omega)$. The degeneracy occurs at the interior point $x=0$. The primary objective is to establish a quantitative weak unique continuation result: if the solution $\phi$ vanishes on a subdomain $\omega \times (0, T)$ (where $\omega$ is an annular region not containing the origin), then $\phi$ must vanish identically on the entire domain $\Omega$.

Methodology
The authors employ a three-step methodology centered on approximation and Carleman estimates:

1. Approximation of the Degenerate Operator:
   Unlike previous works that utilized $A_p$ weights which may lack partial derivatives, this paper constructs a specific $C^{2,1}$ approximation function $\psi_\varepsilon(x)$ for $|x|$. This function coincides with $|x|$ outside a small ball $B_\varepsilon$ and is smoothed within $B_\varepsilon$ to ensure differentiability. The degenerate weight $w = |x|^\alpha$ is approximated by $w_\varepsilon = (\psi_\varepsilon)^\alpha$. This allows the degenerate equation to be approximated by a sequence of non-degenerate parabolic equations with weights $w_\varepsilon$.

2. Derivation of Carleman Estimates:
   The authors derive Carleman estimates for the non-degenerate approximating equations. A key technical contribution is the partitioning of the estimation process into three distinct parts to handle the approximation error near the degenerate point:

   • The region outside the small ball ($Q_\varepsilon^c$), where the weight behaves like $|x|^\alpha$.
   • The region inside the small ball ($Q_\varepsilon$), where the smoothed weight $\psi_\varepsilon$ is used.
   • The boundary terms and the interaction between these regions.
     The estimates are established for the approximate equations and then passed to the limit as $\varepsilon \to 0$ (or $k \to \infty$ where $\varepsilon = 1/k$).

3. Limiting Process and Cut-off Techniques:
   Using the compactness of the solution spaces and the convergence properties of the approximations (Theorem 3.1 and Corollary 3.1), the Carleman estimates for the non-degenerate equations are transferred to the original degenerate equation. Additionally, a cut-off method is employed to isolate the degenerate part of the domain, allowing the application of classical Carleman estimates on the non-degenerate annular regions.

Key Contributions

• Differentiable Approximation of Weights: The paper introduces a novel approximation approach for the $A_p$ weight $|x|^\alpha$. By constructing a $C^{2,1}$ function $\psi_\varepsilon$, the authors ensure the approximate weights possess the necessary partial derivatives required for the rigorous derivation of Carleman estimates, addressing a limitation in previous literature where approximate weights were merely Lebesgue measurable.
• Quantitative Estimates on Annular Domains: The authors establish quantitative weak unique continuation inequalities specifically on annular domains ($\Omega = A_{3R, 6R}$). The results provide explicit bounds relating the $L^2$ norm of the solution in the interior to its norm on a sub-annulus.
• Handling Interior Degeneracy: The work extends unique continuation results to parabolic equations with degeneracy at an interior point ($x=0$) in dimensions $N \ge 2$, a case distinct from the extensively studied $N=1$ scenario.

Main Results
The paper establishes the following quantitative inequality (Theorem 5.3): For a solution $\phi$ of the backward degenerate parabolic equation, there exists a constant $C > 0$ such that:
$$\iint_{\Omega \times (T/4, 3T/4)} |x|^{2-\alpha} |\phi|^2 \, dx \, dt \le C \iint_{\omega \times (0, T)} |\phi|^2 \, dx \, dt,$$
where $\omega$ is an annular subdomain.

From this inequality, the Weak Unique Continuation Property is deduced (Corollary 5.1): If $\phi = 0$ on $\omega \times (0, T)$, then $\phi(0) = 0$ on $\Omega$. By the backward uniqueness of the equation, this implies $\phi \equiv 0$ on $\Omega$.

Significance and Scope
The paper claims significance in providing a rigorous framework for unique continuation in degenerate parabolic equations with interior degeneracy using approximation methods that preserve differentiability. The authors explicitly note the modesty of their scope: the results are currently restricted to annular domains. Extending these quantitative unique continuation results to arbitrary domains is identified as future work. The methodology builds upon and refines approximation techniques previously used for degenerate elliptic equations, adapting them to the parabolic setting with specific attention to the time-dependent nature of the problem.