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On the solutions to variable-order fractional p-Laplacian evolution equation with L^1-data

Sixuan Liu, Gang Dong, Hui Bi, Boying Wu

TL;DR

The paper analyzes Dirichlet-boundaried nonlinear parabolic equations driven by a variable-order fractional $p$-Laplacian $(-\Delta)_p^{s(\cdot,\cdot)}$ with nonnegative $L^1$ data. It develops a well-posedness theory for weak, renormalized, and entropy solutions using approximation, energy estimates, and nonlinear semigroup methods. It proves existence and uniqueness of renormalized and entropy solutions and establishes a comparison principle, with an equivalence between the two solution concepts. The results extend the study of variable-order fractional operators to evolution problems with $L^1$ data and Dirichlet boundaries, providing a rigorous framework for nonlinear nonlocal parabolic equations with singular data.

Abstract

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative $L^1$-data, which has a variable-order fractional $p$-Laplacian operator. The existence and uniqueness of renormalized solutions and entropy solutions to the equation is proved. To address the significant challenges encountered during this process, we use approximation and energy methods. In the process of proving, the well-posedness of weak solutions to the problem has been established initially, while also establishing a comparative result of solutions.

On the solutions to variable-order fractional p-Laplacian evolution equation with L^1-data

TL;DR

The paper analyzes Dirichlet-boundaried nonlinear parabolic equations driven by a variable-order fractional -Laplacian with nonnegative data. It develops a well-posedness theory for weak, renormalized, and entropy solutions using approximation, energy estimates, and nonlinear semigroup methods. It proves existence and uniqueness of renormalized and entropy solutions and establishes a comparison principle, with an equivalence between the two solution concepts. The results extend the study of variable-order fractional operators to evolution problems with data and Dirichlet boundaries, providing a rigorous framework for nonlinear nonlocal parabolic equations with singular data.

Abstract

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative -data, which has a variable-order fractional -Laplacian operator. The existence and uniqueness of renormalized solutions and entropy solutions to the equation is proved. To address the significant challenges encountered during this process, we use approximation and energy methods. In the process of proving, the well-posedness of weak solutions to the problem has been established initially, while also establishing a comparative result of solutions.
Paper Structure (4 sections, 9 theorems, 126 equations)

This paper contains 4 sections, 9 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries and main result
  3. Auxiliary Theorem
  4. Proof of main results

Key Result

Lemma 2.1

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $m(\Omega)\subset\mathbb{R}^{N}$ be a measurable set with finite measure. Fix $x\in\mathbb{R}^{N}$, let $p>1,\;s(x,y)\in(0,1)$, there holds for a suitable constant $c=c(N, p, s^+)>0$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • ...and 13 more