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Asymptotic solution of the conserved phase field system in the fast relaxation case. (English) Zbl 0923.35082

This paper is concerned with the conserved phase field system \[ \begin{cases} \partial_t(\theta+ \ell\phi)= k\Delta\theta+ f(x,t),\\ -\tau_0\partial_t\phi= \xi^2\Delta(\xi^2\Delta\phi+ {1\over 2a} (\phi- \phi^3)+ k_1\theta),\end{cases}\quad (x,t)\in Q,\tag{1} \] where \(Q= \Omega\times (0,T)\), \(\Omega\subset \mathbb{R}^n\) \((n\leq 3)\) is a bounded domain with smooth \((C^\infty)\) boundary \(\partial\Omega\), \(T<\infty\); \(\theta\) is the normalized temperature, \(\phi\) is the transition function, \(f(x,t)\) is a given function, and \(\ell\), \(k\), \(k_1\), \(\tau_0\), \(\xi\), \(a\) are positive parameters.
An asymptotic solution for the conserved phase field system (1) related to the case \(a= \varepsilon/2\), \(\xi=\sqrt\varepsilon\), \(\tau_0= k\varepsilon\), \(k=\text{const}>0\) with natural initial and boundary conditions \(\theta|_{t= 0}= \theta^0(x,\varepsilon)\), \(\phi|_{t=0}= \phi^0(x,\varepsilon)\), \(\partial_N\phi|_\Sigma= \partial_N\phi|_\Sigma= \partial_N\Delta\phi|_\Sigma= 0\) (\(\partial_N= \partial/\partial_N\) – the external normal to \(\partial\Omega\), \(\Sigma= [0,T]\times \partial\Omega\)), describing the free interface dynamics, is constructed and justified. The main result of the paper is the derivation of the limiting problem in \(C(\overline Q)\) as \(\varepsilon\to 0\) for a sequence of tanh-type solutions of the basic mathematical model considered here.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B25 Singular perturbations in context of PDEs
80A22 Stefan problems, phase changes, etc.
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