### A 2D climate energy balance model coupled with a 3D deep ocean model.

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We consider parabolic equations with nonlocal coefficients obtained from the Vlasov-Fokker-Planck equations with potentials. This class of equations includes the classical Debye system from electrochemistry as well as an evolution model of self-attracting clusters under friction and fluctuations. The local in time existence of solutions to these equations (with no-flux boundary conditions) and properties of stationary solutions are studied.

In this paper, a phase field system of Penrose–Fife type with non–conserved order parameter is considered. A class of time–discrete schemes for an initial–boundary value problem for this phase–field system is presented. In three space dimensions, convergence is proved and an error estimate linear with respect to the time–step size is derived.

The aim of this paper is to demonstrate how the variational equations from can be formulated and solved in some abstract Banach spaces without any a priori construction of special linearization schemes. This should be useful e.g. in the analysis of heat conduction problems and modelling of flow in porous media.