Global regular solvability of a nonuniformly nonlinear sixth order Cahn–Hilliard system
The aim of this paper is to prove existence, uniqueness and continuous dependence on the initial datum of a global regular solution (for arbitrarily large time) for a nonuniformly nonlinear sixth order Cahn–Hilliard system. The nonuniform nonlinearity means that the unknown enters nonlinearly the highest 6th order operator. Such nonlinearity results from the second order Ginzburg–Landau (GL) functional with the coefficients of the first and second gradient depending on the solution. The physical motivation of our system comes from the microemulsion model proposed by Gompper and Schick (1989) and Schmid and Schick (1993). However, other applications are expected to be relevant as well. We note that in a particular case our system is the same as the isotropic version of a modified phase-field crystal model. The mathematical treatment of the problem requires elaborate techniques extending our (2011) existence result on a sixth order Cahn–Hilliard system based on a GL functional with constant coefficient of the second order gradient term.