By D Daners

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

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Additional resources for Abstract Evolution Equations, Periodic Problems and Applications

Example text

This function is called the K-functional. Furthermore, for each x ∈ E0 , the function K(· , x; E) is increasing and concave. It is not difficult to prove that K(t, · ; E) t>0 is a family of norms on E0 which are all equivalent to · 0 . Let now θ ∈ (0, 1) and 1 ≤ p < ∞ and define for each x ∈ E0 the expressions x θ,p ∞ := t −θ K(t, x; E) p 0 1 p dt t , and x θ,∞ := sup t−θ K(t, x; E). 1 Definition For each θ ∈ (0, 1) and 1 ≤ p ≤ ∞ set FR θ,p (E) := (E0 , E1 )θ,p := {x ∈ E0 ; x θ,p < ∞}. With this definition we have the following important theorem.

The evolution operator in interpolation spaces We are now ready to derive the crucial estimates for the evolution operator in interpolation spaces. The advantage of prefering to work in these spaces rather than in fractional power spaces, will be evident not only from the simplicity of the proofs, but also from the fact that most of the estimates for the evolution operator are simpler than the corresponding estimates in fractional power spaces, allowing a more natural and elegant treatment of semilinear equations.

1. 11). Strong continuity was proved in Step 1. 33) [t → wt ] ∈ C 1 0, ∞), L1 (Rn ) and w˙ t (x) = 1 n |x|2 − + wt (x). 33), that d (wt ∗ u) = w˙ t ∗ u ∈ X, that is U (t)u ∈ D(AX ) [t → wt ∗ u] ∈ C 1 (0, ∞), X and that dt for all u ∈ X. It remains to prove (2′ ). We can write t w˙ t = − n wt +gt 2 with gt (x) := |x|2 wt . 4t Using polar coordinates and the Γ-function, it follows that gt t w˙ t L1 L1 = n 2 and thus ≤n holds for all t > 0. Since tAX U (t)u = t w˙ t ∗ u, the assertion follows. F. Diagonal operators: Suppose that X 1 , .