Degenerate parabolic equations in divergence form: fundamental solution and Gaussian bounds

1. Problem Statement and Context

The paper investigates second-order degenerate parabolic operators of the form:
$$H u := \partial_t u - \omega^{-1} \text{div}_x (A(t, x) \nabla_x u)$$
defined on $\mathbb{R} \times \mathbb{R}^n$. The key features of the problem are:

• Degeneracy: The ellipticity is not uniform but dictated by a spatial weight $\omega(x)$. Specifically, $\omega^{-1}A$ satisfies the classical uniform ellipticity condition.
• Weight: The weight $\omega$ belongs to the Muckenhoupt class $A_2(\mathbb{R}^n)$, which ensures specific doubling and reverse Hölder properties.
• Coefficients: The matrix $A(t, x)$ has complex, measurable, and time-dependent coefficients.
• Goal: The primary objective is to establish the existence of a fundamental solution (heat kernel) $\Gamma(t, x; s, y)$ and to derive Gaussian upper and lower bounds for this kernel.

2. Methodology

The author employs a combination of functional analysis, semigroup theory, and regularity theory for weak solutions:

• Functional Framework: The problem is set in weighted Sobolev spaces $H^1_\omega(\mathbb{R}^n)$ and $L^2_\omega(\mathbb{R}^n)$. The operator is treated via sesquilinear forms $B_t(u, v)$ defined on these spaces.
• Existence via [AB25]: The paper relies on a method developed in a previous work by the author and Auscher ([AB25]) to prove the existence and uniqueness of a fundamental solution family $\Gamma(t, s)$ as bounded operators on $L^2_\omega$. This approach avoids assuming smoothness of coefficients.
• Off-Diagonal Estimates: The author derives exponential decay estimates for the fundamental solution between disjoint sets using energy methods and Grönwall's lemma, adapted to the weighted setting.
• Equivalence Proof (The Core Innovation): The paper establishes a rigorous equivalence between:
  1. The existence of Gaussian upper bounds for the fundamental solution.
  2. The validity of Moser's $L^2-L^\infty$ estimates (local boundedness) for local weak solutions of both the forward equation ($Hu=0$) and the adjoint equation ($H^*v=0$).
• Real Coefficients Case: When coefficients are real-valued, the paper utilizes the Harnack inequality (known for degenerate equations with real coefficients) to derive Gaussian lower bounds, bypassing the need for Nash's regularity results in the upper bound derivation.

3. Key Contributions and Results

A. Existence and Uniqueness of the Fundamental Solution

The paper confirms that the operator $H$ possesses a unique fundamental solution family $\Gamma = (\Gamma(t, s))_{t,s \in \mathbb{R}}$. This family acts as the inverse of the parabolic operator and satisfies the Chapman-Kolmogorov identities and adjointness properties.

B. The Equivalence Theorem (Main Result)

Theorem 1.2 is the central contribution. It states that the fundamental solution $\Gamma(t, x; s, y)$ satisfies a pointwise Gaussian upper bound:
$$|\Gamma(t, x; s, y)| \leq \frac{K_0}{\sqrt{\omega(B(x, \sqrt{t-s}))}\sqrt{\omega(B(y, \sqrt{t-s}))}} e^{-k_0 \frac{|x-y|^2}{t-s}}$$
if and only if all local weak solutions to $Hu=0$ and $H^*v=0$ satisfy Moser's $L^2-L^\infty$ estimates.

• Significance: This generalizes the seminal result of Hofmann and Kim (2004) from the unweighted case to the weighted, degenerate case with time-dependent complex coefficients. It removes the requirement for smooth coefficients or Nash-type regularity to prove upper bounds; instead, it links the kernel bounds directly to the local boundedness of solutions.

C. Gaussian Bounds for Real Coefficients

In the special case where $A$ has real-valued coefficients:

1. Upper Bounds: The conditions for Moser's estimates are automatically satisfied (due to known regularity theory for real coefficients), providing a direct proof of Gaussian upper bounds without needing Nash's iteration.
2. Lower Bounds: By combining the upper bounds with the Harnack inequality (valid for real coefficients), the author derives matching Gaussian lower bounds:
   $$\frac{\tilde{K}_0}{\sqrt{\omega(B(x, \sqrt{t-s}))}\sqrt{\omega(B(y, \sqrt{t-s}))}} e^{-\tilde{k}_0 \frac{|x-y|^2}{t-s}} \leq \Gamma(t, x; s, y)$$
   This completes the two-sided Gaussian estimate for the heat kernel in the weighted, real-coefficient setting.

D. Conservation Property

The paper proves that the fundamental solution preserves constants (i.e., $\Gamma(t, s)1 = 1$ in $L^2_{\omega, \text{loc}}$), a crucial property for probabilistic interpretations and physical applications.

4. Technical Highlights

• Weighted Geometry: The Gaussian bounds are adapted to the geometry induced by the weight $\omega$. The volume terms $\omega(B(x, \sqrt{t-s}))$ replace the standard Euclidean volume $(t-s)^{n/2}$, reflecting the degenerate nature of the operator.
• Duality Arguments: The proofs heavily utilize duality between the forward operator $H$ and the backward adjoint $H^*$ to transfer estimates from the solution space to the kernel.
• Caccioppoli Inequalities: Localized energy inequalities (Caccioppoli) are established for weak solutions, serving as the foundation for the Moser iteration and the derivation of $L^\infty$ bounds.

5. Significance and Impact

• Generalization: This work significantly extends the theory of parabolic equations with complex coefficients to the degenerate, weighted setting. Previous results often required time-independence or real coefficients for lower bounds.
• Methodological Shift: By proving the equivalence between Gaussian bounds and Moser's estimates, the paper provides a robust framework for analyzing heat kernels in non-smooth, degenerate environments. It suggests that for complex coefficients, the "hard" part of the problem is establishing the local boundedness of solutions; once that is done, the kernel bounds follow naturally.
• Applications: The results are relevant for:
  • Fractional Powers of Parabolic Operators: As mentioned in the introduction, these operators appear in the study of fractional powers.
  • Anomalous Diffusion: Modeling diffusion processes in heterogeneous media where the diffusion coefficient vanishes or blows up according to a weight.
  • Schrödinger Equations: Analysis of heat kernels for Schrödinger operators with singular potentials.

In summary, Khalid Baadi's paper provides a comprehensive and rigorous treatment of degenerate parabolic equations, establishing a fundamental link between the regularity of weak solutions and the pointwise behavior of their fundamental solutions, with explicit Gaussian bounds in both the complex and real coefficient cases.