In the conductive theory of thermal ignition, there is a practically important region of parameter space where the heat conduction partial differential equation with a nonlinear (Arrhenius) heat generation term displays two attractors. Their basins of attraction (particularly that of the minimal solutions) are of great practical importance. Good estimates for the latter region are obtained by a method of spatial averaging of the nonlinear term which give very small errors for large classes of initial conditions, provided the latter are not too 'spikey'. For 'hot spot' type initial conditions the method still works and gives satisfactory estimates. The bound is focussed on the critical initial energy.