Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana–Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana–Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial–boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.