By Tosio Kato
The current article relies at the Fermi Lectures I gave in may possibly, 1985, at Scuola Normale Superiore, Pisa, within which i mentioned numerous equipment for fixing the Cauchy challenge for summary nonlinear differential equations of evolution variety. 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 functions, between different issues, to combined initial-boundary price difficulties for definite nonlinear partial differential equations, similar to elastodynamic and Schrödinger equations.
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Extra resources for Abstract differential equations and nonlinear mixed problems
2. - Quasi linear equations. 1) of height s + 1 rather than 5 + 2. The relevant assumptions are again a straightforward translation of conditions (Ql) to (Q5) and need not be stated explicitly. 14. - Nonlinear wave equations in Q c R"' As the first major application of the abstract nonlinear theory, we shall solve the nonlinear wave equations in Q c with the Neumann boundary condition. The Dirichlet condition is easier to handle, and will be discussed briefly. We note that this section is in principle independent of the results of sections 10 and 12, where linear wave equations are considered, although we make free use of the material obtained there.
0 The first term on the right tends to zero as n —►oo by hypothesis, since (jp in Ys+\. 11). 9 and the bounded convergence theorem. 13). If T is sufficiently small, this term is cancelled by part of the left member. 13) tends to zero as n —>oo. It remains to show that \\df'^\u^ - i^)||o —> 0. 10) and the results already proved. 6 if t' = T. same argument starting from the initial time T', extending the result to a larger interval [0,T"], and so on. This process ends in a finite number of steps, due to the uniformity in conditions (N1-N4).
2. Under the assumptions on the ajk etc. 4a) has a unique solution u G a r . 3. We have assumed for simplicity that the unknown u is scalar valued. 2 holds without modification when u is N vector-valued. In this case the coefficients ajk, etc. N x N matrix-valued. The ellipticity condition for the ajk should be replaced by the symmetry akj = a ( ^ denoting the transposed matrix) and the strong ellipticity, which requires that (ajk(t, c > 0, for all ^ G R ”*, f G where ( | ) denotes the inner product on R ^ .
Abstract differential equations and nonlinear mixed problems by Tosio Kato