In the present contribution we consider a singular phase field system located
in a smooth and bounded three-dimensional domain. The entropy balance equation
is perturbed by a logarithmic nonlinearity and by the presence of an additional
term involving a possibly nonlocal maximal monotone operator and arising from a
class of sliding mode control problems. The second equation of the system
accounts for the phase dynamics, and it is deduced from a balance law for the
microscopic forces that are responsible for the phase transition process. The
resulting system is highly nonlinear; the main difficulties lie in the
contemporary presence of two nonlinearities, one of which under time
derivative, in the entropy balance equation. Consequently, we are able to prove
only the existence of solutions. To this aim, we will introduce a backward
finite differences scheme and argue on this by proving uniform estimates and
passing to the limit on the time step.Comment: Key words: Phase field system; maximal monotone nonlinearities;
nonlocal terms; initial and boundary value problem; existence of solution
