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On the uniqueness and iterative method for the solving of one nonlinear nonstationary problem with non-local boundary conditions of radiation heat transfer type. (Russian. English summary) Zbl 1240.35223
For the problem \[ \rho \dfrac{\partial u}{\partial t}-\dfrac{\partial}{\partial x_i}\left(a_{ij}(t, x, u)\dfrac{\partial u}{\partial x_j}\right)+b(x, t, u, u_x)=f, \quad (x, t)\in Q=G\times[0, T] \] \[ a_{ij}(t, x, u)\dfrac{\partial u}{\partial x_j}\cos(\overline n, x_i)+d(x, t, u)u+h(u(x, t))=\int_{\partial G}h(u(\xi, t))\varphi(\xi, x, t)d\xi+g, \] \[ (x, t)\in S=S_{T}=\partial G\times [0, T] \] \[ u(x, 0)=u^{0}, \quad x\in G, \] where \(G\subset\mathbb R^{n}(n\geq 2)\) is a bounded domain with boundary \(\partial G\), the author has suggested iterative process for its solving, which is convergent beginning with any initial approximation. This convergence to the unique solution is proved at the condition of the smooth solution existence. On the every step of iteration the solving of some linear initial-boundary value with boundary conditions of third kind is required. The convergence rate estimate is obtained together with a priori estimates needed for iterative method construction. Applications to nonstationary problems of radiation heat transfer are given.

35K20 Initial-boundary value problems for second-order parabolic equations
Full Text: MNR