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A monotonicity formula for a semilinear fractional parabolic equation

Ignacio Bustamante

TL;DR

The paper develops a monotonicity formula for the semilinear fractional parabolic equation $$(\partial_t-\Delta)^s u=|u|^{p-1}u,$$ $0<s<1$, by embedding the problem into a high-dimensional elliptic extension through a parabolic-to-elliptic transformation. Building on the elliptic Lane–Emden monotonicity and the parabolic extension framework, it constructs an almost-monotone energy $\mathcal{E}_n(R)$ for the lifted function $V_n$ and shows convergence to a genuine monotone quantity $\mathcal{E}(R)$ as the lift dimension $n\to\infty$, which in turn yields a monotone functional $\mathcal{J}(t)$. The approach unifies nonlocal parabolic analysis with high-dimensional elliptic techniques, providing a fractional analogue of the Giga–Kohn monotonicity formula and enabling potential classification and regularity results for solutions. The results rely on precise volume-element computations and convergence lemmas to justify limiting procedures and yield an explicit derivative formula for the monotone quantity.

Abstract

By applying a high-dimensional parabolic-to-elliptic transformation, we establish a monotonicity formula for the extension problem of the fractional parabolic semilinear equation $(\partial_t -Δ)^s u = |u|^{p-1}u$, where $0<s<1$. This is an analogous result to the Giga-Kohn monotonicity formula for the equation $\partial_t u - Δu = |u|^{p-1}u.$

A monotonicity formula for a semilinear fractional parabolic equation

TL;DR

The paper develops a monotonicity formula for the semilinear fractional parabolic equation , by embedding the problem into a high-dimensional elliptic extension through a parabolic-to-elliptic transformation. Building on the elliptic Lane–Emden monotonicity and the parabolic extension framework, it constructs an almost-monotone energy for the lifted function and shows convergence to a genuine monotone quantity as the lift dimension , which in turn yields a monotone functional . The approach unifies nonlocal parabolic analysis with high-dimensional elliptic techniques, providing a fractional analogue of the Giga–Kohn monotonicity formula and enabling potential classification and regularity results for solutions. The results rely on precise volume-element computations and convergence lemmas to justify limiting procedures and yield an explicit derivative formula for the monotone quantity.

Abstract

By applying a high-dimensional parabolic-to-elliptic transformation, we establish a monotonicity formula for the extension problem of the fractional parabolic semilinear equation , where . This is an analogous result to the Giga-Kohn monotonicity formula for the equation

Paper Structure

This paper contains 10 sections, 10 theorems, 161 equations.

Table of Contents

  1. introduction
  2. Fractional operators and parabolic-to-elliptic transformations
  3. The fractional Lane-Emden equation.
  4. The fractional heat operator.
  5. Parabolic-to-elliptic transformations
  6. An almost monotonicity formula for $V_n$.
  7. Volume elements and convergence lemmas
  8. Induced volume elements.
  9. Convergence lemmas.
  10. Derivation of the monotonicity formula

Key Result

Theorem 2.1

Let $V(z_0,z) \in C^{2}(\mathbb{R}_{+}^{N+1}) \cap C(\overline{\mathbb{R}_{+}^{N+1}})$, such that $V$ obeys $($derivative lane emden eq$)$ and $($first eq extension lane$)$, and suppose $z_0^a \partial_{z_0} V \in C(\overline{\mathbb{R}_{+}^{N+1}})$. For $R>0$, let Then, $E$ is a non-decreasing function of $R$. Moreover,

Theorems & Definitions (21)

  • Theorem 2.1: Theorem 1.4 in Dvila2017
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Remark 4.1
  • Lemma 4.2
  • ...and 11 more