<p>A non-Gaussian Hardy equation is studied with a non-linearity of Osgood-type growth. A fractional derivative in time is incorporated for the first time in a research of this type. Existence of local and global solutions are established by combining properties of the fundamental solutions together with the parameters of the non-Gaussian process, leading to optimal asymptotic estimates. Additional properties of the fundamental solutions and instantaneous blow-up results are found. The Banach contraction mapping principle is particularly exploited. It is also defined a critical exponent for existence and non-existence of solutions together with a judicious choice of the initial data.</p>