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Global solutions to a Penrose-Fife phase-field model under flux boundary conditions for the inverse temperature. (English) Zbl 0859.35049

The authors study the following initial-boundary value problem: \[ \varphi_t-\Delta\varphi\in F_1'(\varphi)-{F_2(\varphi)\over \theta},\;\rho(\theta)_t+F_2'(\varphi)\varphi_t=-\Delta\Biggl({1\over \theta}\Biggr)+g\quad\text{in }\Omega\times (0,\infty), \]
\[ {\partial\varphi\over \partial n}=0,\;{\partial\over \partial n} \Biggl({1\over \theta}\Biggr)=\theta-\theta_\Gamma\text{ on }\Gamma\times (0,\infty),\;\varphi(\cdot,0)=\varphi_0,\;\theta(\cdot,0)=\theta_0\text{ in }\Omega, \] where \(\Omega\) is a smooth, bounded and open domain of \(\mathbb{R}^n\), \(1\leq n\leq 3\), with boundary \(\Gamma\). These kind of systems arise from the Penrose-Fife approach to model the kinetics of phase transitions. One novelty of the paper is that the correct boundary condition for the temperature field \(\theta\) is assumed (in the previous papers, the linear boundary condition \(-\partial\theta/\partial n= \theta-\theta_\Gamma\) is supposed). Although this boundary condition originates some extra difficulties, the authors prove, under essentially the same hypotheses as in previous papers, the global existence and uniqueness of strong solutions.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35R70 PDEs with multivalued right-hand sides
35K55 Nonlinear parabolic equations
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