By S Kawashima, Taku Yanagisawa

A suite of papers on microlocal research, Fourier research within the advanced area, generalized features and comparable issues. lots of the papers originate from the talks given on the convention "Prospects of Generalized capabilities" (held in November 2001 at RIMS, Kyoto). Reflecting the truth that the papers are devoted to Mitsuo Morimoto, the topics thought of during this booklet are interdisciplinary, simply as Morimoto's works are. The ancient backgrounds of the topics in a few of the papers also are mentioned intensive. therefore, the quantity might be priceless not just to the experts within the fields, but in addition to people who have an interest within the heritage of contemporary arithmetic resembling distributions and hyperfunctions Mathematical features of supersonic stream previous wings, S-X. Chen; the null and international life of ideas to platforms of wave equations with diverse speeds, R. Agemi, okay. Yokoyama; scaling limits for big platforms of interacting debris, okay. Uchiyama; regularity of ideas of preliminary boundary worth difficulties for symmetric hyperbolic platforms with boundary attribute of continuous multiplicity, Y. Yamamoto; at the half-space challenge for the discrete pace version of the Boltzmann equation, S. Ukai; on a decay fee of recommendations to the one-dimensional thermoplastic equations of a part line - linear half, Y. Shibata; bifurcation phenomena for the Duffing equation, a. Matsumura; a few comments at the compactness technique, A.V. Kazhikhov; percolation on fractal lattices - asymptotic behaviour of the correlation size, M. Shinoda. (Part contents)

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M. 8). 5 Estimate of the First Derivatives of the Solutions to Initial Value Problems. 1) which has proved by M. Kovalyov 10. 1). %re'/^ie+'i'))di> rdrds f A',F(s, r e ^ ^ 1 " ' ) ^ where x if f 1 X'(s) — (a cos 0, a sin #) = a e * ^ 9

7). 2)-(l8). Then there exists a positive con stant ea depending on given functions such that the initial value problem (11) 52 has a unique C°°- solution in [0, oo) x R 2 for all e with 0 < e < e0. M. 8). 5 Estimate of the First Derivatives of the Solutions to Initial Value Problems. 1) which has proved by M. Kovalyov 10. 1). %re'/^ie+'i'))di> rdrds f A',F(s, r e ^ ^ 1 " ' ) ^ where x if f 1 X'(s) — (a cos 0, a sin #) = a e * ^ 9

Klainerman9). We apply John-Shatah observations to a system of quasilinear wave equa tions with different speeds in two space dimensions. e. ,dum), and the a(i = 1 . . ,m) are positive constants different from each other. f. Setv = du. 2) that the vector v satisfies the system of first order which is hyperbolic near v = 0: 2 Y^a°(v)dav=0. (1-3) For the concrete expression of aa (v) see section 2. O) € R 2 and C / 0. 4) that the vector to satisfies the system in one space dimension: i a°{w)dtw + £ Qal(w)dsw = 0.