Nitin Kumar Maurya | Searcharxiv

arXiv2026

On fractional $1$-Laplacian evolution equation

We study a nonlocal evolution problem driven by the fractional $1$-Laplacian with a Carathéodory nonlinearity satisfying subcritical growth conditions. We develop a potential well framework to investigate the existence and qualitative behavior of solutions at different initial energy levels. By combining a modified potential well method with Galerkin approximations, we establish the global existence of weak and strong solutions under appropriate conditions on the initial energy and the sign of the associated Nehari functional. To address the singular nature of the fractional $1$-Laplacian, we approximate the problem by a family of fractional $p$-Laplacian equations, derive estimates uniform with respect to $p>1$, and pass to the limit as $p\to 1^{+}$. Furthermore, in the low-dimensional regime $N<2s$, we employ a subdifferential approach to establish the local existence of strong solutions and investigate their qualitative behavior.

math.AP↗

arXiv2025

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

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.

math.AP↗