Laminar forced convection in a circular tube is investigated with a boundary condition of prescribed axially varying wall heat flux, under the assumptions of hydrodynamically developed flow and of negligible axial conduction and viscous dissipation in the fluid. A condition on the asymptotic behaviour of the axial distribution of wall heat flux is established which is fulfilled by power-law varying axial distributions and which guarantees the existence of a thermally developed regime. It is proved that for all the axial distributions which fulfil this condition the asymptotic value reached by the local Nusselt number is 48/11, i.e. the same which holds for a uniform wall heat Aux. For some power-law varying axial distributions of wall heat flux, a finite difference determination of the thermal entrance region is performed. In every numerical solution, the local Nusselt number tends asymptotically to 48/11.