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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 \(t\in [0,T]\).

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
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References:

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