On the exact number of monotone solutions of a simplified Budyko climate model and their different stability

Abstract

We consider a simplified version of the Budyko diffusive energy balance climate model. We obtain the exact number of monotone stationary solutions of the associated discontinuous nonlinear elliptic with absorption. We show that the bifurcation curve, in terms of the solar constant parameter, is S-shaped. We prove the instability of the decreasing part and the stability of the increasing part of the bifurcation curve. In terms of the Budyko climate problem the above results lead to an important qualitative information which is far to be evident and which seems to be new in the mathematical literature on climate models. We prove that if the solar constant is represented by λ ∈ (λ1, λ2), for suitable λ1 < λ2, then there are exactly two stationary solutions giving rise to a free boundary (i.e. generating two symmetric polar ice caps: North and South ones) and a third solution corresponding to a totally ice covered Earth. Moreover, we prove that the solution with smaller polar ice caps is stable and the one with bigger ice caps is unstable.

The physics of climate variability and climate change

“…The 'S'-shaped curve in the figure arises from two back-to-back saddle-node bifurcations. Bensid and Diaz (2019) have recently provided a mathematically rigorous treatment of the bifurcation diagram of a Budyko-type 1-D EBM.…”

Doubly nonlinear equation involving p(x)-homogeneous operators: Local existence, uniqueness and global behaviour

“…involving p-Laplacian operator, we refer to [23,34,40,42]. The nonhomogeneous variant of the model (1.1) together with multivalued source/sink terms can also be interpreted as the limiting case (when m → 1) of the climate Energy Balance Models (see [15,16,22]). Recently, the study of D.N.E.…”

Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions

“…It is quite interesting, challenging and very difficult to study the structure of the global attractors for this type of inclusions. Nevertheless, some progress has been done already in the papers [2,4,5].…”