An initial-boundary-value problem for the sixth order Cahn-Hilliard type equation in 3-D is studied. The problem describes phase transition dynamics in ternary oil-water-surfactant systems. It is based on the Landau-Ginzburg theory proposed for such systems by G. Gompper et al. We prove that the problem under consideration is well posed in the sense that it admits a unique global smooth solution which depends continuously on the initial datum.

An initial-boundary-value problem for a class of sixth order viscous Cahn-Hilliard type equations with a nonlinear diffusion is considered. The study is motivated by phase-field modelling of various spatial structures, for example arising in oil-water-surfactant mixtures and in modelling of crystal growth on atomic length, known as phase field crystal model. For such problem we prove the existence and uniqueness of a global in time regular solution. First the finite-time existence is proved by means of the Leray-Schauder fixed point theorem. Then, due to suitable estimates, the finite-time solution is extended step by step on the infinite time interval.

A two-dimensional thermoviscoelastic system of Kelvin–Voigt type with strong dependence on temperature is considered. The existence and uniqueness of a global regular solution is proved without small data assumptions. The global existence is proved i

A three-dimensional thermo-visco-elastic system for Kelvin-Voigt type material at small strains is considered. The system involves nonlinear temperature-dependent specific heat relevant in the limit of low temperature range. The existence of a unique global regular solution is proved without small data assumptions. The proof consists of two parts. First the existence of a local in time solution is proved by the Banach successive approximations method. Then a lower bound on temperature and global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time. The paper generalizes the results of the previous author's publication in SIAM J. Math. Anal. 45, No. 4 (2013), pp. 1997–2045.