By Tosio Kato

The current article relies at the Fermi Lectures I gave in might, 1985, at Scuola Normale Superiore, Pisa, during which i mentioned a variety of equipment for fixing the Cauchy challenge for summary nonlinear differential equations of evolution style. the following I current a close exposition of 1 of those equipment, which offers with “elliptic-hyperbolic” equations within the summary shape and which has purposes, between different issues, to combined initial-boundary price difficulties for convinced nonlinear partial differential equations, equivalent to elastodynamic and Schrödinger equations.

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**Example text**

This matches the need for the application of the contraction map theorem, which works from 0-norm to itself since there is no elliptic equation to be solved. 3. - The main theorem. We want to solve (Q) for u{t) for a given г¿(0) = . Again 0 has to satisfy the compatibility condition. To formulate the latter, we have to compute the initial set 0o = (^ = w(0), = ^¿г¿(0), ... , 0s = 5fw(0) of the unknown u. 10) have to be modified, since here v = u rather than v = dtu and W c Ys rather than W C Fs+i.

8a) |9(aj[i;u,u]| < c||u||v||u||y. 8b) ||at/l,(i)w||v. < c||u||y. 8), \dt(f\9)t\ = I < f , A M - ' ( d t A M ) U t ) - ' g > \ < c '| | / | | y . | | f f | | y . , where we have used the fact that As(t) is an isomorphism of V onto V*. This proves that ( | )i is Lipschitz continuous in t. 8b) also shows that (iii') is true for the symmetric part As of A. 8a) for too. Condition (ii') is met because D{A{t)) = V = H". 4. 1. 9) a[t\u,v] = j (ajk(t,x){dku)djV + ak(t,x)(dku)v Q + x)udjV + a(i, x)uv)dx + J 6(i, x)uvdS^ r where (ajkit^x)) is a real symmetric, positive definite matrix with lower bound > 6 > 0.

0 5 be the associated initial set determined above. 13) 5[iy(0) = 0r, r = 0, . . ,s , t e I, where i? > 0 is a constant, sufficiently small that the ball with center and radius R is contained in W . e. t. 13) is a serious question. We have not been able to prove it under assumptions (N0) to (N4), although there should be no difficulty in verifying it in most applications (cf. [BF]). Thus we shall add another assumption: (N5) E^{I) is nonempty for some R > 0 and / = [0 ,T ], T > 0. Of course if E^(I) is nonempty, E^(I') is nonempty for I' = [0, T'], T' < T.

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