An Energy Balance Climate Model with Hysteresis
by
Georg Hetzer
Auburn University

It is well-known that the advance of ice-sheets at the beginning of an ice-age occurs at a rate which is considerably slower than the rate by which they retreat during the transition to an interglacial period. In order to account for such an effect in an energy balance climate model, one introduces a hysteresis functional for the latent energy flux density of ice formation. This leads to a functional reaction-diffusion problem of the form

ì ï ï ï ï í ï ï ï ï î

\partialt[c(x)u +w]- Ñ·[k(x)|Ñu|p-2Ñu] +g(u)=f(t, x, u, w)       x in M,  t > 0 w(t, x)= Wx[v(·, x)](t) ,  t > 0,  x in M, v(t, x):= ó õ 0 -T  \beta(s, x)u(t+s, x) ds,  t > 0,  x in M, u(s, x)= \vartheta(s, x),  -T <= s <= 0, x in M,

on the two-sphere M with linear or slow diffusion (p >= 2). Global existence and boundedness of solutions were obtained for a corresponding problem without hysteresis in a joint joint paper with J.I. Díaz. An extension of these results will be presented here in case that W is a Duhem hysteresis operator.

Date received: March 26, 1999