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

Khalid Baadi

TL;DR

This work analyzes second-order degenerate parabolic equations in divergence form with a spatial $A_2$-weight, linking the existence of a generalized fundamental solution with Gaussian upper bounds to Moser’s $L^2$-$L^{\infty}$ estimates for local weak solutions. By developing a weighted Lions-type embedding, a robust Cauchy problem framework, and off-diagonal decay, the authors establish a sharp equivalence between kernel bounds and solution regularity, including a clear treatment of real-valued coefficients that yields Gaussian lower bounds via Harnack and Nash-type regularity. The results extend previous unweighted and time-independent-coefficient theories to the weighted, possibly time-dependent, and complex-coefficient setting, removing smoothness assumptions on $A$ while preserving Gaussian-type control. Applications touch on fractional parabolic diffusion, anomalous diffusion, and heat kernels in weighted contexts, providing a foundational toolkit for analyzing degenerate parabolic processes in divergence form with measurable coefficients.

Abstract

In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial $A_2$-weight. We prove that having a generalized fundamental solution with upper Gaussian bounds is equivalent to Moser's $L^2$-$L^\infty$ estimates for local weak solutions. In the special case of real coefficients, Moser's $L^2$-$L^\infty$ estimates are known, which provide an easier proof of Gaussian upper bounds, and a known Harnack inequality is then used to derive Gaussian lower bounds.

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

TL;DR

This work analyzes second-order degenerate parabolic equations in divergence form with a spatial -weight, linking the existence of a generalized fundamental solution with Gaussian upper bounds to Moser’s - estimates for local weak solutions. By developing a weighted Lions-type embedding, a robust Cauchy problem framework, and off-diagonal decay, the authors establish a sharp equivalence between kernel bounds and solution regularity, including a clear treatment of real-valued coefficients that yields Gaussian lower bounds via Harnack and Nash-type regularity. The results extend previous unweighted and time-independent-coefficient theories to the weighted, possibly time-dependent, and complex-coefficient setting, removing smoothness assumptions on while preserving Gaussian-type control. Applications touch on fractional parabolic diffusion, anomalous diffusion, and heat kernels in weighted contexts, providing a foundational toolkit for analyzing degenerate parabolic processes in divergence form with measurable coefficients.

Abstract

In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial -weight. We prove that having a generalized fundamental solution with upper Gaussian bounds is equivalent to Moser's - estimates for local weak solutions. In the special case of real coefficients, Moser's - estimates are known, which provide an easier proof of Gaussian upper bounds, and a known Harnack inequality is then used to derive Gaussian lower bounds.

Paper Structure

This paper contains 23 sections, 22 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and basic assumptions
  3. The weight and function spaces
  4. The distributional duality bracket
  5. The degenerate parabolic operator
  6. Existence, uniqueness, representation and regularity results
  7. A Lions' type embedding with integral identities
  8. Fundamental solution
  9. The Cauchy problem and the fundamental solution
  10. Further properties arising from the divergence form
  11. $L^2$-decay for the fundamental solution
  12. Local weak solutions and Caccioppoli inequality
  13. Local weak solutions
  14. Caccioppoli inequality
  15. Conservation property
  16. ...and 8 more sections

Key Result

Theorem 1.1

The operator $\mathcal{H}=\partial_t - \omega^{-1}\mathrm{div}_x (A(t,\cdot) \nabla_x )$ has a unique fundamental solution $\Gamma=(\Gamma(t,s))_{t,s \in \mathbb{R}}$. Moreover, the following properties hold.

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: The degenerate parabolic operator
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.4: Fundamental solution for $\mathcal{H}=\partial_t - \omega^{-1}\mathrm{div}_x(A(t,\cdot) \nabla_x)$
  • Theorem 3.5
  • proof
  • Proposition 3.6: Cauchy problem on $(0,\mathfrak{T})$
  • ...and 42 more