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On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions

Irene Gonzálvez, Fernando Quirós, Fernando Soria, Zoran Vondraček

TL;DR

This work develops a comprehensive Widder-type framework for nonlocal heat equations driven by general Lévy-type operators. It proves uniqueness for nonnegative and very weak solutions with prescribed initial traces, establishes existence of initial traces under admissible growth, and constructs representation formulas via heat kernels, encompassing purely nonlocal, anisotropic, and mixed local/nonlocal operators. The theory leverages heat-kernel estimates, kernel comparability, and Bernstein-type representations (notably for fractional Laplacians) to obtain regularity, trace, and existence results, with explicit anisotropic constructions and a local/nonlocal extension. The results provide a unified, flexible toolkit for analyzing nonlocal diffusion, initial-data traces, and solution representations across a broad class of operators, with clear pathways to concrete examples such as anisotropic sums of fractional Laplacians and operators with a local part.

Abstract

We develop a Widder-type theory for nonlocal heat equations involving quite general Lévy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal Lévy operators defined through positive symmetric Lévy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part.

On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions

TL;DR

This work develops a comprehensive Widder-type framework for nonlocal heat equations driven by general Lévy-type operators. It proves uniqueness for nonnegative and very weak solutions with prescribed initial traces, establishes existence of initial traces under admissible growth, and constructs representation formulas via heat kernels, encompassing purely nonlocal, anisotropic, and mixed local/nonlocal operators. The theory leverages heat-kernel estimates, kernel comparability, and Bernstein-type representations (notably for fractional Laplacians) to obtain regularity, trace, and existence results, with explicit anisotropic constructions and a local/nonlocal extension. The results provide a unified, flexible toolkit for analyzing nonlocal diffusion, initial-data traces, and solution representations across a broad class of operators, with clear pathways to concrete examples such as anisotropic sums of fractional Laplacians and operators with a local part.

Abstract

We develop a Widder-type theory for nonlocal heat equations involving quite general Lévy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal Lévy operators defined through positive symmetric Lévy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part.

Paper Structure

This paper contains 38 sections, 79 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. The heat kernel
  3. Lévy processes and heat kernels
  4. Heat kernel properties for operators L_K
  5. Completely monotone functions and fractional heat kernels
  6. Operators L_K
  7. Very weak solutions
  8. An alternative definition of very weak solution
  9. Uniqueness of very weak solutions
  10. Existence of an initial trace
  11. Existence of very weak solutions
  12. Classical solutions
  13. Uniqueness of classical solutions
  14. Existence of classical solutions
  15. General purely nonlocal operators
  16. ...and 23 more sections

Key Result

Proposition 2.1

Let $m$ be the Fourier symbol of $\mathcal{L}_K$, with $K$ satisfying assumptions A0--A2. Then, $m\in C(\mathbb{R}^d)$ and there exist constants $0<C_{1}\leq C_{2}$ such that where $0<\beta_{1}\le\beta_{2}$ are the exponents in the global weak scaling conditions A2.

Theorems & Definitions (136)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 126 more