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Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {ℝ}^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(ℝ;L²\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value $u_0\in H^1\left(\Omega \right)$ is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when $u_0\in L^2\left(\Omega \right)$ and the integral kernel in the nonlocal boundary condition is symmetric.