Tokamak ignition equilibria and their stability are examined in a radially dependent model with empirical (Alcator) scaling for the energy confinement time. The empirical energy loss term corresponds to a diffusion operator, whose stationary eigenfunctions are the ignition equilibria. The dispersion relation for fluctuations around these equilibria is then derived, yielding the threshold for thermal instability. The instability growth rate is non-linearly enhanced by broadening of the temperature profile, which broadening is itself caused by an increase in temperature. Saturation of the instability occurs at a lower temperature than that indicated by zero-dimensional considerations. The model system appears to have no built-in saturation mechanism, other than the D-T reaction slow-down.