The author considers the semilinear singular parabolic problem \[ L_ ku=f(x,u)\quad on\quad \Omega:=(0,1)\times (0,T), \] where T is a positive number, f a regular function and, for k real: \[ L_ ku:=u_{xx}+(k/x)u_ x-u_ t. \] The above parabolic equation is supplemented with initial conditions and boundary conditions as follows: let \(\Phi\) (x,t) be a given function: then it is required that \(u(x,0)=\Phi (x,0)\) on \(B_ 0:=(0,1)\times \{0\},\) and, denoting \(\{\) (r,t): \(0\leq t<T\}\) by \(S_ r\), either \(u(x,t)=\Phi (x,t)\) on \(S_ 1\cup S_ 0\) (problem I), or \(u(x,t)=\Phi (x,t)\) on \(S_ 1\) (problem II).
In case of problem I, the solution is sought in the space \[ C^ 2(\Omega)\cap C(S_ 0\cup S_ 1\cup B_ 0), \] while for problem II the solution should belong to \[ C^ 2(\Omega)\cap C(\Omega \cup S_ 1\cup B_ 0). \] The author proves, by a suitable approximation procedure, that problem II has a solution for any real k, while problem I has a solution whenever \(k<1\). The proofs make (non-trivial) use of comparison techniques, whereby the notion of barrier function plays a crucial role.