On the solutions to variable-order fractional <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian evolution equation with <inline-formula><tex-math id="M2">$ L^1 $</tex-math></inl

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.