A geometric correction in the problem on the motion of a free boundary. (English. Russian original) Zbl 0922.35191

Omel’yanov, G. A.; Trushkov, V. V.

Math. Notes 63, No. 1, 137-139 (1998); translation from Mat. Zametki 63, No. 1, 151-153 (1998).

We consturct a correction to the asymptotic solution of the following system of phase field equations [G. Caginalp, Arch. Ration. Mech. Anal. 92, 205-245 (1986; Zbl 0608.35080)]: \[ \partial_t(\theta+ \varphi)=\Delta\theta,\quad t\in(0,T),\quad x\in\Omega,\quad \varepsilon^2\partial_t\varphi= \varepsilon^2\Delta\varphi+ \varphi- \varphi^3+ \varepsilon\kappa\theta, \]
\[ \theta|_\Sigma= g(x,t)|_\Sigma,\quad \varphi|_\Sigma= 1,\quad \theta|_{t=0}= \theta_0(x,\varepsilon),\quad \varphi|_{t= 0}= \varphi_0(x,\varepsilon). \] Here \(\Sigma= \partial\Omega\times (0,T)\), \(\Omega\) is a domain in \(\mathbb{R}^n\) with smooth \((C^\infty)\) boundary, \(\varepsilon\) is a small parameter, \(\kappa>0\) is a constant, and \(\partial_t= \partial/\partial t\).

MSC:

35R35	Free boundary problems for PDEs
35Q72	Other PDE from mechanics (MSC2000)
80A22	Stefan problems, phase changes, etc.

Keywords:

self-similar asymptotic solutions of kink type; mean curvature; Gibbs-Thomson condition; asymptotic solution; phase field equations

Citations:

Zbl 0608.35080

References:

[1]	G. Caginalp,Arch. Rational Mech. Anal.,92, 205–245 (1986). · Zbl 0608.35080 · doi:10.1007/BF00254827
[2]	V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich,Differentsial’nye Uravneniya [Differential Equations],31, No. 3, 483–491 (1995).
[3]	V. P. Maslov and G. A. Omel’yanov,Uspekhi Mat. Nauk [Russian Math. Surveys],36, No. 3, 63–126 (1981).
[4]	V. P. Maslov, V. G. Danilov, and K. A. Volosov,Mathematical Modeling of Mass and Heat Transfer Processes [in Russian], Nauka, Moscow (1987). · Zbl 0645.73049
[5]	A. A. Lacey and A. B. Taylor,IMA J. Appl. Math.,30, 303–314 (1983). · Zbl 0531.35076 · doi:10.1093/imamat/30.3.303
[6]	V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich,Free Boundary Problem News,6, 27–28 (1995).