A qualitative analysis of a second-order anisotropic phase-field transition system endowed with a general class of nonlinear dynamic boundary conditions

Abstract

The paper is concerned with the study of a nonlinear second-order anisotropic phase-field transition system of Caginalp type, subject to nonlinear and in-homogeneous dynamic boundary conditions (in both unknown functions). Under certain hypothesis on the input data: $ f_{_1}(t,x) $, $ f_{_2}(t,x) $, $ w_{_1}(t,x) $, $ w_{_2}(t,x) $, $ u_0(x) $, $ \alpha_0(x) $, $ \varphi_0(x) $ and $ \xi_0(x) $, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $ W^{1,2}_p(Q)\times W^{1,2}_p(\Sigma) $, $ W^{1,2}_\nu(Q)\times W^{1,2}_p(\Sigma) $. Here we extend the previous results concerned with nonlinearity of cubic type, allowing to the present mathematical model to be more capable for describing the complexity of a wide class of real physical phenomena (moving interface problems, image processing, the phase changes at the boundary $ \partial\Omega $, etc.).