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By D Daners

A part of the "Pitman learn Notes in arithmetic" sequence, this article covers: linear evolution equations of parabolic sort; semilinear evolution equations of parabolic style; evolution equations and positivity; semilinear periodic evolution equations; and purposes.

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This kind of situation arises for instance when dealing with initial boundary value problems where the boundary operator depends on time. For a brief description of such a generalization see the Notes and References at the end of Section 5. 3. Interpolation spaces In this section we review those concepts of interpolation theory which are necessary for our purposes. Our intention is to apply interpolation theory rather than to develop it. 38 This means that we will not dwell at length in the abstract theory, developing it as far it is convenient for our purposes.

The same proof also workes for X0 = BU C(Rn ). 10 of that paper. Since in the case of the Laplacian this theorem remains valid for α ∈ (1/2, 1), we see that the asserion is also true for such α’s. 19 Corollary Let α ∈ (0, 1) \ {1/2}. 4 in the appendix. E. Interpolation and fractional powers: In the introduction we mentioned the fact that it has been common practice to work in the so called fractional power spaces rather than in the interpolation space setting we shall adopt in this work. In this subsection, we shall briefly explain the relationship between these two types of spaces.

They supply us with examples of admissible families. We will also give examples of some concrete interpolation spaces associated to certain function spaces arising in the applications. We emphasize that everything we need from abstract interpolation theory is contained above. Notes and references: Some standard references in interpolation theory are Bergh and L¨ ofstr¨ om [22], Triebel [122] or Krein, Petunin and Semenov [85]. Further books dealing with certain aspects of interpolation theory are [26], [31].

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