Variational Techniques for a One-Dimensional Energy Balance Model

Abstract. A one-dimensional climate energy balance model (1D-EBM) is a simplified climate model that describes the evolution of Earth's temperature based on the planet's energy budget. In this study, we examine a 1D-EBM that incorporates a bifurcation parameter representing the impact of carbon dioxide on the energy balance. Firstly, independent of the value of the additive parameter, we demonstrate the existence of a steady-state solution by solving the associated variational problem, which involves minimizing a potential functional. Again using variational techniques, we can give sufficient conditions to prove the existence of at least three-steady state solutions. Secondly, we establish the uniqueness of the solution for the variational problem by examining the differentiability of the value function, which represents the minimum value of the potential functional across all temperature profiles. Lastly, we explore how this characterization provides valuable insights into the structure of the bifurcation diagram. Specifically, we demonstrate a one-to-one correspondence between the derivative of the value function and the mean value of the minimizer for the variational problem. Furthermore, we show the applicability of our findings to more general reaction-diffusion spatially heterogeneous models.