Inserted: 1 jul 2017
Last Updated: 6 aug 2018
Journal: Nonlinear Anal. Real World Appl.
In the present contribution we consider a singular phase field system located in a smooth bounded 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 is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear and 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 nite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step.
Keywords: Phase field system, Maximal monotone nonlinearities, Initial and boundary value problem