The initial-to-final inverse problem for the heat operator

Technical Summary: The Initial-to-Final Inverse Problem for the Heat Operator

Problem Statement
This paper addresses the inverse problem of determining a heat-generation coefficient $V(t, x)$ in the heat equation from the "initial-to-final map." The physical model considers a region $\mathbb{R}^n$ ($n \ge 2$) where thermal conduction occurs, and the medium generates internal heat at a rate proportional to the heat density $u(t, x)$, governed by the equation:
$$\partial_t u - \Delta u = V u \quad \text{in } (0, T) \times \mathbb{R}^n,$$
with an initial condition $u(0, \cdot) = f$.

The forward problem defines a bounded linear map $U_T: L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, which assigns to every initial heat density $f$ its final heat density $u(T, \cdot)$. The inverse problem asks: under what conditions on $V$ can the coefficient be uniquely determined solely from the knowledge of the map $U_T$?

The authors focus on coefficients $V \in L^1((0, T); L^\infty(\mathbb{R}^n))$ that exhibit super-exponential decay, meaning $e^{\rho|x|}V \in L^\infty(\Sigma)$ for all $\rho > 0$, where $\Sigma = (0, T) \times \mathbb{R}^n$.

Methodology
The approach adapts the strategy previously developed by Caro and Ruiz for the Schrödinger equation to the parabolic setting. The methodology proceeds through four main stages:

1. Integral Identity (Orthogonality Relation):
   The authors first establish that if two coefficients $V_1$ and $V_2$ yield the same initial-to-final map ($U_T^1 = U_T^2$), then for any solution $u_1$ of the forward equation with potential $V_1$ and any solution $v_2$ of the adjoint (final-value) equation with potential $V_2$, the following orthogonality relation holds:
   $$\int_\Sigma (V_1 - V_2) u_1 v_2 \, dx dt = 0.$$

2. Construction of Exponentially Growing Solutions:
   To utilize this relation, the authors construct special solutions that grow exponentially in space and time. They seek solutions of the form:
   $$u = e^{|\nu|^2 t + \nu \cdot x} (u^\sharp + u^\flat),$$
   where $\nu \in \mathbb{R}^n$ is a large parameter.

   • Leading Term ($u^\sharp$): This term is chosen to be constant in the direction of $\nu$ and solves the heat equation in the hyperplane orthogonal to $\nu$. It is constructed using the heat kernel restricted to this hyperplane.
   • Correction Term ($u^\flat$): This term accounts for the perturbation caused by $V$. The construction requires solving an integral equation involving a specific operator $S_\nu$. A major technical hurdle is ensuring that $u^\flat$ remains negligible compared to the leading term as $|\nu| \to \infty$.

3. Key Estimate via Gaussian Moments:
   The core technical contribution lies in proving that the correction term $u^\flat$ decays sufficiently fast. The authors define a weighted $L^2$ space and prove a crucial estimate for the operator $S_\nu$ (Proposition 5.1). This estimate relies on the moment generating function of a Gaussian random variable (Lemma 5.1), which allows them to control the translation invariance of the heat kernel in the direction of $\nu$. Specifically, they show that for large $|\nu|$, the norm of the correction term scales as $O(1/|\nu|)$.

4. Extension of the Orthogonality Relation:
   Since the constructed solutions $u^\nu$ and $v^\nu$ grow exponentially, they do not belong to the standard solution space $C([0, T]; L^2(\mathbb{R}^n))$. The authors rigorously extend the orthogonality relation (2) to these exponentially growing solutions by exploiting the parabolic regularity of the heat equation and using cut-off functions to justify integration by parts (Section 6).

Key Results
The main result is Theorem 1, which states:
Consider $V_1, V_2 \in L^1((0, T); L^\infty(\mathbb{R}^n))$ with $n \ge 2$. If both $V_1$ and $V_2$ have super-exponential decay, then:
$$U_T^1 = U_T^2 \implies V_1 = V_2.$$
The proof proceeds by substituting the exponentially growing solutions into the extended orthogonality relation. As $|\nu| \to \infty$, the contribution of the correction terms vanishes, leaving an integral involving only the leading terms $u^\sharp$ and $v^\sharp$. By varying the parameters in the leading terms, the authors reduce the problem to a Fourier transform argument, ultimately showing that the difference $V_1 - V_2$ must be zero almost everywhere.

Significance and Claims
The paper claims to initiate the study of the initial-to-final inverse problem specifically for the heat equation. While the general structure parallels the Schrödinger case studied in [8], the authors highlight that the parabolic nature of the heat operator introduces distinct challenges and opportunities.

• Extension of Theory: The work extends the uniqueness results for initial-to-final state maps from the Schrödinger equation to the parabolic setting.
• Technical Innovation: The authors emphasize their derivation of a weighted $L^2$-estimate for the correction terms, which is a novel adaptation of Gaussian moment generating functions to the heat operator context. This estimate is essential for handling the non-compact nature of the perturbations in $\mathbb{R}^n$.
• Distinction from Boundary Data: Unlike the extensive literature on inverse problems for parabolic equations using boundary measurements (e.g., Dirichlet-to-Neumann maps), this work addresses the recovery of the zeroth-order coefficient from the initial-to-final map. This formulation allows for the consideration of non-compact perturbations that might be beyond the scope of standard boundary data approaches.

The paper does not propose new experimental applications or future directions beyond establishing this theoretical uniqueness framework. It positions itself as a foundational step in understanding data-driven prediction for heat diffusion systems, analogous to the quantum mechanical motivation provided for the Schrödinger equation.