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On variable Lebesgue spaces and generalized nonlinear heat equations

Gastón Vergara-Hermosilla

TL;DR

This work studies the Cauchy problem for a generalized nonlinear heat equation with fractional diffusion in variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. By leveraging decay estimates of the fractional heat kernel and the functional-analytic structure of variable exponent spaces (including log-Hölder continuity, maximal operators, and Riesz potentials), the authors establish two well-posedness results via a contraction mapping: a global mild solution in the mixed space $\mathcal{L}^{p(\cdot)}_{\frac{nb}{2α-\langle 1 \rangle_γ}}(\mathbb{R}^n,L^\infty_t)$ for small data, and a local mild solution in $L^{p(\cdot)}([0,T],L^q(\mathbb{R}^n))$ using a new class of embedding exponents $\overline{q}(\cdot)$. The analysis handles nonlinearities $F(u)=|u|^b u$ and convective forms, and demonstrates the first applications of variable Lebesgue spaces to these generalized nonlinear heat equations. The results provide a flexible framework for studying evolution PDEs with spatially varying integrability, extending classical Lebesgue-space methods. They also highlight how fractional diffusion and variable exponent techniques can yield precise global or local well-posedness under smallness or embedding conditions. Overall, the paper broadens the toolbox for nonlinear diffusion problems by integrating variable exponent theory with fractional-derivative dynamics.

Abstract

In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations. In a first theorem, we prove the existence and uniqueness of global-in-time mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{ \frac{nb}{2α- \langle 1 \rangle_γ} } (\mathbb{R}^n,L^\infty([0,T[ ))$. On the other hand, by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$.

On variable Lebesgue spaces and generalized nonlinear heat equations

TL;DR

This work studies the Cauchy problem for a generalized nonlinear heat equation with fractional diffusion in variable Lebesgue spaces . By leveraging decay estimates of the fractional heat kernel and the functional-analytic structure of variable exponent spaces (including log-Hölder continuity, maximal operators, and Riesz potentials), the authors establish two well-posedness results via a contraction mapping: a global mild solution in the mixed space for small data, and a local mild solution in using a new class of embedding exponents . The analysis handles nonlinearities and convective forms, and demonstrates the first applications of variable Lebesgue spaces to these generalized nonlinear heat equations. The results provide a flexible framework for studying evolution PDEs with spatially varying integrability, extending classical Lebesgue-space methods. They also highlight how fractional diffusion and variable exponent techniques can yield precise global or local well-posedness under smallness or embedding conditions. Overall, the paper broadens the toolbox for nonlinear diffusion problems by integrating variable exponent theory with fractional-derivative dynamics.

Abstract

In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces . More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations. In a first theorem, we prove the existence and uniqueness of global-in-time mild solutions in the mixed-space . On the other hand, by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution in the space .
Paper Structure (16 sections, 20 theorems, 76 equations)

This paper contains 16 sections, 20 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. General setting
  3. Presentation of the results
  4. Preliminaries
  5. Variable Lebesgue spaces
  6. Some key estimates on the fractional heat kernel
  7. Mild solutions in variable Lebesgue spaces
  8. Proof of Theorem \ref{['Theoreme_2']}
  9. Proof.
  10. Proof.
  11. Proof.
  12. Proof of Theorem \ref{['Theoreme_1']}
  13. Proof.
  14. Proof.
  15. Proof.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\gamma \in \{0,1\}$ fixed. Consider $\alpha\in ]\frac{1}{2}, 1]$, a variable exponent $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^n)$ such that $p^->1$, an initial data $u_0\in \mathcal{L}^{p(\cdot)}_{ \frac{nb}{2\alpha - \langle 1 \rangle_\gamma } }(\mathbb{R}^n)$, and let $f$ be an exte

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1: Contraction mapping principle
  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.1
  • Lemma 2.2
  • Definition 2.2
  • Lemma 2.3
  • ...and 23 more