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The mapping f-+j is an injection; we will identify j and f Furthermore, if 45 4. 43) U'(O, T;X) c: ~'(]O, T[;X). 0 which defines f(k)E~'(]O, T[;X). Furthermore, the mapping f~Pk) is linear continuous from ~'(]O, T [; X) into itself. 45) Xc: Y. 47) dv/dtE~'(]O, ~'(]O, T [; X), hence in particular T[; Y). 48) dv/dtEU(O, T; y) (1 ~q~ (0). Then the function v is continuous from [0, T]~ Y, possibly after a modification on a set of measure zero. This is not the best possible result 21 . We will state the result more precisely for a particular pair X, Y-a pair we will use very frequently in the remainder ofthe book.

25) with v=v(x) and integrate by parts over Q; let a/on be the normal derivative on r directed toward the exterior of Q. 27) Sr(ou(t)/on +

(A)=kA, k>O, the problem is non-linear. 4. 32) (see graph in Fig. 9); "'(A) = J~ 4>(Jl)dJl ,, ,, , ,, , , ,, , / / "',J91) (92V '" , ,, , '" o Fig. 6), and is of type 1. 1. 7. 2. 5. Let", be given by (Fig. 34) "'(A)= I tk(A-h)2, 0, A~h , A~h .

3. Spaces of Vector-Valued Functions For a precise formulation of the problems of evolution posed in Section 3, we need additional tools which we will now introduce. 40) IlfIILOO(O,T;X) = esssup Ilf(t)llx· te(O,T) Ilflb(o,T;x) < 00 (p# (0), This is a Banach space. 41) ~'(]O, T[;X) = 2'(~(]0, T[;X) where, generally, 2'(Y; X) denotes the space of linear continuous mappings of Y-+X. 42) j(cp) = J~ f(t)cp(t)dt, ([0, T]), (a well defined mapping cp-+j(cp), linear continuous from ~(]O, TD-+X). The mapping f-+j is an injection; we will identify j and f Furthermore, if 45 4.