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| We study a nonlocal version of the two-phase Stefan problem, which models a phase transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in deriving a theory for sign-changing solutions of the equation, $u_t=J\ast v -v $, $v=\Gamma(u)$, where the monotone graph is given by $\Gamma(s)={\mathop{\rm sign}}(s)(|s|-1)_+$. We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behaviour for sign-changing solutions, which present challenging difficulties due to the non-monotone evolution of each phase.\\
References:\\
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$[2]$~ Br\"andle, C.; Chasseigne, E.; Quir\'os,
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one-phase Stefan model, SIAM J. Math. Anal., Vol. 44, No. 4,
(2012) 3071--3100.\\
$[3]$~ Meirmanov, A. M. ``The Stefan problem''. Walter de Gruyter, Berlin, 1992.\\
$[4]$~ Stefan, J., \"Uber einige Probleme der Theorie
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(1889), 473-484; see also pp. 614-634; 965-983; 1418-1442. |
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