Global Existence and Finite-Time Blow-Up of Solutions for Parabolic Equations Involving the Fractional Musielak $g_{x,y}$-Laplacian

TL;DR

The paper addresses global existence and finite-time blow-up for parabolic equations driven by the fractional Musielak $g_{x,y}$-Laplacian in bounded domains. It develops a comprehensive variational-analytic framework in fractional Musielak–Orlicz–Sobolev spaces, employing a modified potential-well method and Galerkin approximations to obtain local existence of strong solutions and to classify long-time behavior by energy thresholds. The main contributions include local well-posedness via a subdifferential approach, detailed global existence and blow-up results across low, critical, and high initial energy regimes, and explicit analyses of the unstable/stable regions with respect to the Nehari functional and the well depth $d$. The results extend nonlocal models with variable growth and provide a unifying approach for a wide class of operators, with potential applications in image processing and materials science. The work also discusses open questions, such as extending beyond the $\Delta_2$-condition and establishing $s\to1$ limits, highlighting directions for future research.

Abstract

In this work, we study the parabolic fractional Musielak $g_{x,y}$-Laplacian equation: \begin{equation*} \left\{ \begin{aligned} u_{t} + (-Δ)_{{g}_{x,y}}^{s} u &= f(x,u), && \text{in } Ω\times (0, \infty), u &= 0, && \text{on } \mathbb{R}^N \setminus Ω\times (0, \infty), u(x,0) &= u_0(x), && \text{in } Ω, \end{aligned} \right. \end{equation*} where $(-Δ)_{{g}_{x,y}}^{s}$ denotes the fractional Musielak $g_{x,y}$-Laplacian, and $f$ is a Carathéodory function satisfying subcritical growth conditions. Using the modified potential well method and Galerkin's method, we establish results on the local and global existence of weak and strong solutions, as well as finite-time blow-up, depending on the initial energy level (low, critical, or high). Moreover, we explore a class of nonlocal operators to highlight the broad applicability of our approach. This study contributes to the developing theory of fractional Musielak-Sobolev spaces, a field that has received limited attention in the literature. To our knowledge, this is the first work addressing the parabolic fractional $g_{x,y}$-Laplacian equation.

Theorems & Definitions (76)

• Remark 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6
• Lemma 2.7
• Definition 2.8
• Remark 2.9