On convergence of implicit difference scheme for a nonlinear equation of nonstationary filtration type (Q1897910)

An implicit difference scheme for the nonlinear equation of nonstationary filtration type \[ \partial \varphi (u)/\varphi t + Lu = f,\quad Lu = - \sum^n_{i = 1} \partial (\alpha_i(x, u)k_i (x, \nabla u))/\partial x_i \tag{1} \] is considered. Here \(\varphi (\xi)\) is a monotonically increasing function, its behavior near both origin and infinity is like that of \(|\xi|^{\alpha - 2}\), \(\alpha > 1\); the space operator \(L\) maps \(\overset \circ {W}^{(1)}_p (\Omega)\) into \(W^{-1}_{p'} (\Omega)\), \((p > 1)\). It is continuous, coercive, bounded; its monotonicity is not necessary, but the authors assume that it satisfies a certain condition which generalizes the monotonicity concept. Three theorems on the convergence of complements of solutions of the implicit difference scheme to the generalised solution of the differential problem are proved. The first theorem is proved under minimal restrictions on the initial data and under sufficiently general assumptions concerning the function \(\varphi\) and the operator \(L\), but under rather restrictive conditions on the parameters \(\alpha\) and \(p\). In the two other theorems the convergence of the solutions of the implicit difference scheme and the existence of the generalized solution of equation (1) for arbitrary \(\alpha\) and \(p\), but under additional suppositions concerning the operator \(L\), the function \(\varphi\), and the smoothness of the initial data, are proved.