Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.10634 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.