Abstract parabolic initial boundary value problems with singular data and with values in interpolation spaces

We consider abstract initial boundary value problems for parabolic differential-operator equations on the rectangle [0, T] × [0, 1] with singular data. We use our previous results on norm-estimates of solutions and R-boundedness of some sets of boundary value problems for abstract elliptic equations with a parameter on [0, 1] in a UMD Banach space. Unique solvability of these problems is proved in the Sobolev spaces of vector-valued functions with values in some interpolation spaces. The corresponding estimates for the solutions are also established. We also show completeness of elementary solutions of abstract parabolic boundary value problems. Abstract results are provided by a relevant application to parabolic PDEs. In some cases, the boundary conditions may contain the intermediate points of the interval [0, 1] or may be integro-differential.