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Continuous dependence for a nonstandard Cahn-Hilliard system with nonlinear atom mobility. (English) Zbl 1292.35297

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.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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