By Takaaki Nishida, etc., M. Mimura, H. Fujii

Half I of this quantity surveys the advancements within the research of nonlinear phenomena in Japan up to now decade, whereas half II involves updated unique papers relating qualitative theories and their purposes. handled listed here are nonlinear difficulties regarding normal research, fluid dynamics, mathematical biology and machine sciences, and their underlying mathematical constructions, e.g. nonlinear waves and propagations, bifurcation phenomena, chaotic phenomena, and fractals. the amount is devoted to Professor Masaya Yamaguti in occasion of his sixtieth birthday.

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**Extra info for Patterns and Waves: Qualitative Analysis of Nonlinear Differential Equations**

**Example text**

It may be assumed that SZ is a neighborhood of the origin and the initial hypersurface is {xo=O}. , PK=f, do~ul,o=o=q5j (j=O,. - - m-1) , , where q5, and f are given functions and u is unknown. 1. One of them is that the solution has the regularity in proportion to one of the data and the coefficients of equation. The other is the finite propagation speed (the existence of a sharp region for the N. IWASAKI 18 uniqueness). The case in this section also has these properties, while its special feature is that it may happen that the grade of loss in the regularity grows in proportion to the size of the subprincipal symbol.

Of the Taniguchi Symposium on hyperbolic equations and related topics, Kinokuniya Publisher, Tokyo, 1986, 99-110. [16] --, Applications of Nash-Moser theory to nonlinear Cauchy problems, to appear in Proc. Sym. , 45 (1986). [17] -, The Cauchy problem for hyperbolic equations with double characteristics, Publ. , 19 (1983), 927-942. [18] K. Kajitani, Cauchy problem for non-strictly hyperbolic systems, Publ. , 15 (1979), 519-550. [19] K. Kasahara and M. Yamaguti, Strongly hyperbolic system of linear partial differential equations with constant coefficients, Mem.

For all u of V,n H,", for all g of HIPtzmand for all a : O g a l p , l '(K: g)11a 2 ' R ( Ilglldtm+ 11sI l m l l u l l a + m ) and ll@(' ( u)* ': 9)I l R ca(llg I I R t m l l @ ( u ) Ilm f llgllmll@(u) I I a t m + llgllnlIl'(u)ll 71LIIullrZt9ft) * 34 N. ) I a n d 0 to one s from HOPtzm to HIP Theorem (Nash-Moser). (existence) W e assume the existence of a right parametrix. Let parameters satisfy a relation pfm>r>2a>6m. there exists Then for any open set V , of V , and for any bounded ball B, of HOT, a positive E such that it holds that for all f of @(Vl n B, n H,") Ilf Ill(a-m> 2~ or that an element u of V,n Ho(@exists and satisfies @(u)=O, where Ho(p)= H,B.