Weak solvability and well-posedness of some fractional parabolic problems with vanishing initial datum

This paper tackles a class of nonlinear parabolic equations driven by the fractional p-Laplacian operator with a vanishing initial datum. Our primary aim is to investigate the well-posedness (existence and uniqueness) of solutions to the proposed model. Notably, we will establish two interesting results concerning the existence and uniqueness of weak solutions. The first result pertains to the scenario where the source term is independent of the solution. In this case, we demonstrate the existence and uniqueness of the solution via the classical monotone operator theory modulus vanishing initial datum. The second result deals with the case where the source term is nonlinear and strongly dependent on the solution. To establish the existence of a weak solution in this scenario, we will rely essentially on the use of Schaefer’s fixed point theorem and supplement our approach with some new technical estimates.