  A subdiffusion problem in which the diffusion term is related to a stable
stochastic process is introduced. Linear models of these systems have been
studied in a general way, but non-linear models require a more specific
analysis. The model presented in this work corresponds to a non-linear
evolution equation with fractional time derivative and a pseudo-differential
operator acting on the spatial variable. This type of equations has a couple of
fundamental solutions, whose estimates for the L_p-norm are found to obtain
three main results concerning mild and global solutions. The existence and
uniqueness of a mild solution is based on the conditions required in some
parameters, one of which is the order of stability of the stochastic process.
The existence and uniqueness of a global solution is found for the case of
small initial conditions and another for non-negative initial conditions. The
relationship between Fourier analysis and Markov processes, together with the
theory of fixed points in Banach spaces, is particularly exploited. In
addition, the present work includes the asymptotic behavior of global solutions
as a linear combination of the fundamental solutions with L_p-decay.
