Hölder space solutions of free boundary problems that arise in combustion theory (Q2786978)

In this paper, the author studies multidimensional one-phase free boun\-dary problems for the heat equation. The unique solvability of four such problems is established in Hölder spaces for small times and coercive estimates are obtained for the solutions.NEWLINENEWLINETo describe the four problems at hand more precisely, let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with boundary \(\Gamma\) and let \(\Omega_k(t), t \in [0,T]\), be domains with boundary \(\gamma_k(t)\) such that \(\gamma_k(0)=\Gamma\) and \(\Omega_k(0)=\Omega, k=1,\ldots, 4\). We also let \(N:\Gamma \rightarrow \mathbb{R}^n\) be a unit vector field on \(\Gamma\).NEWLINENEWLINEThe main focus of this paper is the question of unique solvability for the following four problems, in which one is required to find the function \(u_k(x,t)\) and the free boundary \(\gamma_k(t), k=1,\ldots 4\):NEWLINENEWLINE{Problem 1:} NEWLINE\[NEWLINE \begin{aligned} \partial_t u_1 -a\Delta u_1 & = f_1(x,t) \text{ in } \Omega_1(t)\times (0,T), \\ \gamma_1(t)|_{t=0} & = \Gamma, \quad u_1|_{t=0}=u_{01}(x) \text{ in } \Omega, \\ u_1 & =0, \;|\nabla u_1|=\varphi_1(x,t), \;x\in \gamma_1(t), \;t\in (0,T). \end{aligned}NEWLINE\]NEWLINE {Problem 2:} NEWLINE\[NEWLINE\begin{aligned} \partial_t u_2-a\Delta u_2 & = f_2(x,t) \text{ in } \Omega_2(t)\times (0,T), \\ \gamma_2|_{t=0} & =\Gamma, \quad u_2|_{t=0}=u_{02}(x) \text{ in } \Omega, \\ u_2 & =0, \;|\nabla u_2|=-\sigma(x,t)V_N+\varphi_2(x,t), \;x\in \gamma_2(t), \;t\in (0,T). \end{aligned}NEWLINE\]NEWLINE {Problem 3:} NEWLINE\[NEWLINE \begin{aligned} \partial_t u_3 - a\Delta u_3 & =0 \text{ in } \Omega_3(t)\times(0,T), \\ \gamma_3|_{t=0} & =\Gamma, \quad u_3|_{t=0}=u_{03}(x) \text{ in } \Omega, \\ u_3 & =0, \;|\nabla u_3|=c_0, \;x\in \gamma_3(t), \;t\in (0,T). \end{aligned}NEWLINE\]NEWLINE {Problem 4:} NEWLINE\[NEWLINE\begin{aligned} \partial_tu_4-a\Delta u_4 & =0 \text{ in } \Omega_4(t)\times (0,T), \\ \gamma_4(t)|_{t=0} & =\Gamma, \quad u_4|_{t=0}=u_{04}(x) \text{ in } \Omega, \\u_4 & =0, \;|\nabla u_4|=c_0-\sigma(x,t)V_N, \;x\in \gamma_4(t), \;t\in (0,T). \end{aligned}NEWLINE\]NEWLINE Here, \(a>0\) and \(c_0\) are constants and \(V_N\) is the velocity of the movement of the free boundary along \(N(\xi)\).NEWLINENEWLINEThe main results of this paper are the unique solvability of these problems under certain conditions on \(N\), \(\Gamma\), \(\sigma\), \(u_{0k}, k=1,\ldots 4\), \(f_j\) and \(\varphi_j\), \(j=1,2\).