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

TL;DR

This paper establishes existence and uniqueness of solutions for a nonlinear parabolic equation involving a variable-order fractional p-Laplacian with L^1-data, using approximation and energy methods.

Contribution

It introduces novel methods to prove well-posedness for a complex variable-order fractional p-Laplacian evolution equation with L^1-data.

Findings

Proved existence of renormalized and entropy solutions.

Established uniqueness and comparison principles.

Demonstrated well-posedness of weak solutions.

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.