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Summary  An initial value problem for the following singular parabolic differential equation     $$u_{x\alpha } + \frac{{2v}}{x}u_x = u_t (2v > - 1)$$       is studied in some suitable weighted Sobolev class. Such an equation is related to axially symmetric problems and also occurs in probability theory. Uniqueness, existence and representation theorems for the solution are proved and differentiability and regularity properties are investigated. The non-homogeneous equation is also considered.

On a singular parabolic equation related to axially symmetric heat potentials