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On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms

Khalid Baadi

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive variational framework to study well-posedness of degenerate parabolic equations with time-dependent, possibly unbounded lower-order terms in weighted spaces. It introduces a robust set of function spaces built from the degenerate Laplacian -Δ_ω, proves invertibility and causality under subcritical smallness conditions, and constructs a fundamental solution with a representation formula. The results yield L^2 off-diagonal estimates and Gaussian upper bounds for the fundamental solution, extending classical unweighted theory to weighted, degenerate contexts and accommodating mixed Lebesgue and Lorentz lower-order terms. The framework also encompasses inhomogeneous and Lorentz-space extensions, and includes precise energy estimates and regularity conclusions for weak solutions, with potential applications to parabolic PDEs in weighted media.

Abstract

In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded lower-order terms. The unbounded lower-order terms are allowed to lie in mixed time-space Lebesgue or even Lorentz spaces. Our notion of weak solutions is formulated under minimal assumptions. We prove the existence and uniqueness of a fundamental solution, which coincides with the associated evolution family for the homogeneous problem (i.e., with zero source term) and provides a representation formula for all weak solutions. We also establish $L^2$ off-diagonal estimates for the fundamental solution and derive Gaussian upper bounds under the weak assumption of Moser's $L^2$-$L^\infty$ estimates for weak solutions. Our approach is purely variational and avoids any a priori regularity assumptions on weak solutions or regularization via smooth approximations. Two key ingredients are norm inequalities for fractional powers of the degenerate Laplacian, and a set of embeddings that ensure time continuity of weak solutions, extending the classical Lions regularity theorem and accommodating a wide class of source terms.

On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive variational framework to study well-posedness of degenerate parabolic equations with time-dependent, possibly unbounded lower-order terms in weighted spaces. It introduces a robust set of function spaces built from the degenerate Laplacian -Δ_ω, proves invertibility and causality under subcritical smallness conditions, and constructs a fundamental solution with a representation formula. The results yield L^2 off-diagonal estimates and Gaussian upper bounds for the fundamental solution, extending classical unweighted theory to weighted, degenerate contexts and accommodating mixed Lebesgue and Lorentz lower-order terms. The framework also encompasses inhomogeneous and Lorentz-space extensions, and includes precise energy estimates and regularity conclusions for weak solutions, with potential applications to parabolic PDEs in weighted media.

Abstract

In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded lower-order terms. The unbounded lower-order terms are allowed to lie in mixed time-space Lebesgue or even Lorentz spaces. Our notion of weak solutions is formulated under minimal assumptions. We prove the existence and uniqueness of a fundamental solution, which coincides with the associated evolution family for the homogeneous problem (i.e., with zero source term) and provides a representation formula for all weak solutions. We also establish off-diagonal estimates for the fundamental solution and derive Gaussian upper bounds under the weak assumption of Moser's - estimates for weak solutions. Our approach is purely variational and avoids any a priori regularity assumptions on weak solutions or regularization via smooth approximations. Two key ingredients are norm inequalities for fractional powers of the degenerate Laplacian, and a set of embeddings that ensure time continuity of weak solutions, extending the classical Lions regularity theorem and accommodating a wide class of source terms.

Paper Structure

This paper contains 29 sections, 36 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Basic Assumptions, Preliminaries, and Parabolic Tools
  3. Muckenhoupt and reverse Hölder weight classes
  4. Function spaces and the degenerate Laplacian
  5. The distributional anti-duality bracket
  6. Parabolic embeddings and integral identities
  7. Lower-order coefficients in mixed Lebesgue spaces
  8. The variational space and an identification
  9. Embeddings
  10. The degenerate parabolic operator dtplusB
  11. The variational approach
  12. Invertibility and causality
  13. Existence and uniqueness results
  14. Uniqueness
  15. Source term in suitable subspaces of $\dot{V}_0^\star$
  16. ...and 14 more sections

Key Result

Theorem 1.1

There exists a constant $\varepsilon_0 = \varepsilon_0(M, \nu, [\omega]_{A_2}, [\omega]_{RH_{\frac{q}{2}}}, n, q) > 0$ such that, if $P_{r, q} \leq \varepsilon_0$, then the following assertions hold.

Theorems & Definitions (95)

  • Theorem 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4: The distributional bracket
  • Remark 2.5
  • Proposition 2.6: Various homogeneous embeddings
  • Proposition 2.7: Integral identities
  • Lemma 3.1: Linking $L^2(I; D_{S,-1})$ and $L^2(I; \dot{H}^{-1}_\omega(\mathbb{R}^n))$
  • proof
  • ...and 85 more