Alternative methods are used to set conditions and investigate, in the space L1(R3 × R+ mdmdx) the well-posedness of a fractional fragmentation process in a kinetic-diffusive-wave medium. In the analysis, three separate models of diffusion are studied. Techniques like separation of variables and subordination principle are used to finally prove that the Cauchy problem for fractional fragmentation dynamics in a kinetic-diffusive-wave system is well-posed and admits a solution operator that is positive and contractive. This work brings a contribution that may lead to the full explanation of strange phenomena like shattering and sudden appearance of an infinite number of particles in a system that occur in the dynamics of fragmentation process and which remain partially unsolved.