Conditions for the existence of a classical solution of a cauchy type problem for the diffusion equation with a Riemann-Liouville partial derivative

Abstract

We study a Cauchy type problem for a differential equation containing a fractional Riemann-Liouville partial derivative of order α, 0 < α < 2. Conditions under which the solution of the problem tends to zero as |x| → ∞ are obtained. We prove an existence theorem for a classical solution of the Cauchy type problem and show that the solution has a singularity as t → 0 of order 1 − α if 0 < α ≤ 1 and of order 2 − α if 1 < α < 2.