  Osgood functions in the source term are used to produce results for
non-existence of local solutions into the framework of non-Gaussian diffusion
equations. The critical exponent for non-existence of local solutions is found
to depend on the fractional derivative, the non-Gaussian diffusion and the
non-linear term. The instantaneous blow-up phenomenon is studied by exploiting
estimates of the fundamental solutions. Nevertheless, theory of super-solutions
and fixed points are combined for showing existence of global solutions. In
this case, the critical exponent for existence of global solutions depends only
on the last two parameters above.
