The well-posedness of difference parabolic problems is investigated. The following abstract initial value problem with a parameter \(h\) is considered: \[ {dy \over dt} + A_h y = f_h, \quad t \in \mathbb{R}, \quad y(0) = y_{0h}, \tag{1} \] where \(A_h\) is a linear bounded operator acting on a Banach space \(E_h\), \(f_h\) is a given \(E_h\)-valued function, \(y_{0h} \in E_h\) and \(y = y(t)\) is an \(E_h\)-valued function interpreted as a solution of (1). A general difference schemes corresponding to (1) may be put in the following canonical form \[ y(t_{k+1})= [I - \tau U_{\tau h}] y(t_k) + \tau F_{\tau k} (t_k) \quad k = 0,1,\dots;\;y(t_0) = y_{0h}, \tag{2} \] with suitable linear operator \(U_{\tau h}\) and function \(F_{\tau h}\), where \(\tau > 0\) is a mesh stepsize and \(t_k = k\tau\), \(k = 0,1,\dots\).
The main result concerns the well-posedness of the scheme (2) under the hypotheses that the operator \(A_h\) is uniformly almost sectorial of a natural power on \(E_h\) and a discretization method leading to (2) is \(A\)-stable. It is shown that some Runge-Kutta methods (like the Radau methods and the Lobatto methods) and the ones studied by Butcher, Vinokurov and Iuvchenko satisfy the above hypotheses. The main result is applied to the analysis of the difference schemes for multidimensional parabolic problems considered in the mesh analogues of the Lebesgue spaces.