Backward Euler method for abstract time-dependent parabolic equations with variable domains

Abstract

We consider semidiscretizations in time, based on the backward Euler method, of an abstract, non-autonomous parabolic initial value problem \(\) where \(A(t) : D(A(t)) \subset X \to X\), \(0 \le t \le T\), is a family of sectorial operators in a Banach space X. The domains \(D(A(t))\) are allowed to depend on t. Our hypotheses are fulfilled for classical parabolic problems in the \(L^p\), \(1 < p <+\infty\), norms. We prove that the semidiscretization is stable in a suitable sense. We get optimal estimates for the error even when non-homogeneous boundary values are considered. In particular, the results are applicable to the analysis of the semidiscretizations of time-dependent parabolic problems under non-homogeneous Neumann boundary conditions.