Poghosyan, M.; Teymurazyan, R. One-phase parabolic free boundary problem in a convex ring. (English. Russian original) Zbl 1187.35293 J. Contemp. Math. Anal., Armen. Acad. Sci. 44, No. 3, 192-204 (2009); translation from Izv. Nats. Akad. Nauk Armen., Mat. 44, No. 3, 67-84 (2009). Let \(\Omega_i\subset\mathbb R^n\times [0,\infty]\), \(i=0,1\), \(\Omega_0\subset \Omega_1\), \(\Omega_i^T:=\Omega_i\cap\{t\leq T\}\), \(i=0,1\), \(\Omega_1(t_0):=\Omega_1\cap\{t=t_0\}\), \(\Gamma_i\) be a lateral boundary of \(\Omega_i\), \(i=0,1\). The domain \(\Omega_0\) expands in time and \(K_0:=\Omega_0\cap \{t=0\}\) is not empty set. It is required to find a domain \(\Omega_1\) and a function \(u(x,t)\) defined in a ring domain as a solution of the problem \(u_t=\Delta u\) in (1) \(\Omega_1 \setminus {\overline \Omega}_0\),(2) \(u=1\) on \(\Gamma_0\),(3) \(u=0\) and \(|\nabla u|=1\) on \(\Gamma_1\),(4) \(u|_{t=0}=u_0(x)\) in \(K_1 \setminus {\overline K}_0\), where \(K_1:= \text{supp}_{x\in \mathbb R^n\setminus K_0} u_0(x)\cup K_0\) is a compact, convex set.The authors establish that there exists \(T_0>0\), such that the problem (1)–(4) has a unique solution \(u \in C^{2,1}_{x t}(\Omega_1^T \setminus {\overline \Omega}_0^T)\cap C (\overline \Omega_1^T\setminus \Omega_0^T)\), a free boundary possesses Lipschitz regularity in \(t\in [0,T]\) and a domain \(\Omega_1(t)\) is a convex and expands in time for \(t\in [0,T]\). Reviewer: Galina Bizhanova (Almaty) Cited in 1 Document MSC: 35R35 Free boundary problems for PDEs 35K05 Heat equation 80A25 Combustion 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35B65 Smoothness and regularity of solutions to PDEs Keywords:parabolic equation; free boundary problem; combustion; classical solution; existence; uniqueness PDFBibTeX XMLCite \textit{M. Poghosyan} and \textit{R. Teymurazyan}, J. Contemp. Math. Anal., Armen. Acad. Sci. 44, No. 3, 192--204 (2009; Zbl 1187.35293); translation from Izv. Nats. Akad. Nauk Armen., Mat. 44, No. 3, 67--84 (2009) Full Text: DOI References: [1] I. Athanasopoulos, L. Caffarelli and S. Salsa, ”Caloric Functions in Lipschitz Domains and the Regularity of Solutions to Phase Transition Problems”, Annals of Math., 143, 413–434 (1996). · Zbl 0853.35049 · doi:10.2307/2118531 [2] J. I. Diaz, B. Kawohl, ”On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings”, J. Math. Anal. Appl. 177(1), 263–286 (1993). · Zbl 0802.35087 · doi:10.1006/jmaa.1993.1257 [3] A. Henrot and H. Shahgholian, ”Existence of Classical Solutions to a Free Boundary Problem for the p-Laplace Operator: (I) the Exterior Case”, J. Reine Angew. Math., 521, 85–97 (2000). · Zbl 0955.35078 [4] A. Henrot and H. Shahgholian, ”Existence of Classical Solutions to a Free Boundary Problem for the p-Laplace Operator: (II) the Interior Case”, Indiana Univ. Math. J., 49(1), 311–323 (2000). · Zbl 0977.35148 · doi:10.1512/iumj.2000.49.1711 [5] A. Petrosyan, ”On Existence and Uniqueness in a Free Boundary Problemfrom Combustion”, Comm. Partial Differential Equations, 27(3–4), 763–789 (2002). · Zbl 0998.35070 · doi:10.1081/PDE-120002873 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.