Existence and uniqueness of $L^1$-solutions to time-fractional nonlinear diffusion equations

TL;DR

The paper addresses the existence and uniqueness of $L^1$-solutions to time-fractional nonlinear diffusion equations, including porous medium and fast diffusion types, on the whole space. It develops an $L^1$-theory via bounded-domain approximations, leveraging nonlocal time derivatives with completely positive kernels and Yosida-type regularizations to obtain $L^1$-contraction and well-posedness. Key contributions include global $L^1$-existence, uniqueness, and contraction for the Cauchy problem, mass conservation for the fast diffusion nonlinearity with $0<m<1$, and nonextinction results in the time-fractional fast diffusion regime, reflecting memory effects. The results extend the classical $L^1$-theory to time-fractional diffusion and provide a rigorous foundation for nonlinear diffusion models with memory.

Abstract

We establish the global existence and uniqueness of $L^1$-solutions to the Cauchy problem for time-fractional porous medium type nonlinear diffusion equations. Furthermore, we give the mass conservation law for $L^1$-solutions to time-fractional fast diffusion equations, and prove that the finite-time extinction does not occur for any nonnegative $L^1$-solutions.