Existence of a global classical solution of one problem arising in combustion theory (Q2784881)

The author considers the following mathematical model of a combustion process: \(\Delta u - u_t = 0, \) \( (x, t) \in \Omega_T \cup G_T, \) where \(\Omega_T = \{(x, t) \in D_T: u(x, t) < 0\}, \) \(G_t = \{(x, t) \in D_T: u(x, t) > 0\}, \) \(D_T = \{x \in {\mathbb{R}}^3: 0 < R_1 < |x|< R_2\} \times (0, T). \) On the free boundary \(\gamma_t=\partial\Omega_T\cap D_T\) the conditions \(u^+=u^-=0,\) \((u^+_\nu)-(u^-_\nu) = Q^2(x),\) where \(\nu\) is the unit normal, are fulfilled. On the known part of the boundary, \(u(x,t)=\varphi(x, t).\) The initial conditions are: \(u(x,0)=\psi(x), \) \( x \in \{ x\in {\mathbb{R}}^3: 0<R_1<|x|<R_2\},\) \(\varphi(x,0)=\psi(x),\) \(x\in\{x\in{\mathbb{R}}^3: |x|<R_1\}\cup \{x \in {\mathbb{R}}^3: |x|<R_2\}.\)NEWLINENEWLINENEWLINEFor proving the existence of a solution the author constructs a differential-difference approximation of the original problem and proves existence of a solution. Then the author proves uniform estimate and performs the limiting transition.