Colli, P.; Gilardi, G.; Podio-Guidugli, P.; Sprekels, J. Continuous dependence for a nonstandard Cahn-Hilliard system with nonlinear atom mobility. (English) Zbl 1292.35297 Rend. Semin. Mat., Univ. Politec. Torino 70, No. 1, 27-52 (2012). This important paper deals with a phase field system that is addressed and investigated in a rather general framework. The initial and boundary problem consists in looking for two fields, the chemical potential \(\mu\) and the order parameter, that solve \[ \begin{aligned}\varepsilon \partial_t \mu + 2 \rho \partial_t \mu + \mu \partial_t \rho - \Delta \mu = 0\,\, &\text{ in}\,\,\Omega \times (0,T),\\ \delta \partial_t \rho - \Delta \rho + f' (\rho) = \mu\,\, &\text{ in}\,\, \Omega \times (0,T),\\ \partial _n \mu = \partial_n \rho = 0\,\, &\text{ in}\,\, \Gamma \times (0,T),\\ \mu (\cdot,0)=\mu_0\,\, \text{and}\,\, \rho (\cdot,0)=\rho_0\,\, &\text{ in}\,\, \Omega.\end{aligned}\tag{1} \] The existence and uniqueness of the solution to the initial boundary value problem (1) with \(\varepsilon >0\), is proved. The asymptotic behavior of such solutions as \(\varepsilon \searrow 0\) is dicussed by showing a suitable convergence to a (weaker) solution of the limiting problem with \(\varepsilon = 0\). The authors show some regularity properties of the solutions and prove continuous dependence of the solution on the initial data. Reviewer: Jan Lovíšek (Bratislava) Cited in 5 Documents MSC: 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs Keywords:chemical potential; ordered parameter; smooth boundary; double-well potential; Cahn-Hilliard equation; residual dissipation inequality; diffusive phase segregation process; microentropy imbalance; microenergy balance; free energy; locally Lipschitz continuous; uniform parabolicity; structural assumptions; Neumann boundary condition; compact embedding; existence; approximation, translation operator; a priori estimate; distributions; regularity; proper Besov space; optimal condition; uniqueness PDFBibTeX XMLCite \textit{P. Colli} et al., Rend. Semin. Mat., Univ. Politec. Torino 70, No. 1, 27--52 (2012; Zbl 1292.35297)