Well‐Posedness of a Semilinear Fractional Diffusion Equation With Caputo Derivative and Special Boundary Conditions

ABSTRACT We study a semilinear time‐fractional diffusion equation in one spatial dimension, where the time derivative is understood in the sense of Caputo. The problem is complemented with suitable boundary conditions, possibly of moving type. We establish the existence, uniqueness, and regularity of weak solutions under natural assumptions on the data and nonlinearities. The analysis relies on fractional calculus tools, including estimates for the Mittag–Leffler function and a fractional Grönwall inequality. Extensions to nonlinear boundary conditions and moving interfaces are also discussed.