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By Tosio Kato

The current article relies at the Fermi Lectures I gave in may well, 1985, at Scuola Normale Superiore, Pisa, within which i mentioned a number of equipment for fixing the Cauchy challenge for summary nonlinear differential equations of evolution sort. right here I current a close exposition of 1 of those equipment, which bargains with “elliptic-hyperbolic” equations within the summary shape and which has functions, between different issues, to combined initial-boundary price difficulties for yes nonlinear partial differential equations, resembling elastodynamic and Schrödinger equations.

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1). CHAPTER II Applications This chapter is intended for applications of the abstract theory developed in Chapter I. The most interesting applications will be to the system of generalized wave equations, linear and nonlinear, in a bounded domain in with the Dirichlet or Neumann boundary condition. ) Such equations will cover many equations in mathematical physics, in particular elastodynamics. Since, however, the wave equations are of second order in time, we find it convenient to develop a variant of the abstract theory for second-order evolution equations.

The following proposition will partially meet this requirement, although it is somewhat complicated. 3. Assume that: (i'a) H and H' have structures o f real Hilbert space, with the inner products ( I )t and ( I ){ depending on t e l , such that the norms induced by them are equivalent to the standard norms || ||// and || \\hi, respectively. These inner products are Lipschitz continuous in t, in the sense that \{u\v)t - (w|i^)s| < c|i - s|||'u||^||'u||^, and similarly for ( | )[. (i'b) {A{t)} is a family o f (unbounded) linear operators in H, such that A{t) = As{t) + Ar(t), where As(t) is selfadjoint and uniformly positive-definite with respect to the inner product ( | )i, while Ar(t) is uniformly bounded (with bound K , say) from H' into H.

1. 1), where we substitute for u a 33 ABSTRACT EVOLUTION EQUATIONS, ETC. given function w{t) G W and seek a solution v. Here the linear theory will be applied with the family A^(t) = A(w(t)) and the suhscale of {X j\Y j} of height s (see section 7). (b) Condition (N1) is rather implicit and complicated. It could be deduced from conditions related to higher order derivatives of F, but this will not necessarily simplify the matter. It seems to us that the implicit form (N1) is more convenient in applications.

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Abstract differential equations and nonlinear mixed problems by Tosio Kato


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