Abstract
We consider initial abstract boundary value problems for parabolic differential-operator equations in UMD Banach spaces settings on the rectangle [0,T]×[0,1]. We use our previous results on norm-estimates of solutions of boundary value problems for abstract elliptic equations with a parameter on[0,1] in a UMD Banach space. Unique solvability of the problems is proved in the spaces of vector-valued continuous functions. The corresponding estimates of the solution are also established. Then, completeness of a system of root functions of abstract elliptic boundary value problems and completeness of elementary solutions of initial abstract parabolic boundary value problems are obtained. All abstract results are provided by a relevant application to parabolic or elliptic PDEs. We also treat, in applications, integro-differential equations and boundary conditions.
| Original language | English |
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| Pages (from-to) | 152-171 |
| Number of pages | 20 |
| Journal | Proceedings of the Institute of Mathematics and Mechanics |
| Volume | 40 |
| Issue number | Special Issue |
| State | Published - 2014 |