By S Kawashima, Taku Yanagisawa

A set of papers on microlocal research, Fourier research within the complicated area, generalized services and comparable issues. many of the papers originate from the talks given on the convention "Prospects of Generalized features" (held in November 2001 at RIMS, Kyoto). Reflecting the truth that the papers are devoted to Mitsuo Morimoto, the themes thought of during this ebook are interdisciplinary, simply as Morimoto's works are. The historic backgrounds of the themes in a number of the papers also are mentioned intensive. hence, the amount will be important not just to the experts within the fields, but in addition to those that have an interest within the heritage of contemporary arithmetic resembling distributions and hyperfunctions Mathematical points of supersonic move earlier wings, S-X. Chen; the null and international life of ideas to structures of wave equations with assorted speeds, R. Agemi, okay. Yokoyama; scaling limits for big platforms of interacting debris, ok. Uchiyama; regularity of ideas of preliminary boundary worth difficulties for symmetric hyperbolic structures with boundary attribute of continuous multiplicity, Y. Yamamoto; at the half-space challenge for the discrete speed version of the Boltzmann equation, S. Ukai; on a decay fee of strategies 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 approach, A.V. Kazhikhov; percolation on fractal lattices - asymptotic behaviour of the correlation size, M. Shinoda. (Part contents)

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12) that the eigenvector £±(0) corresponding to Xf (0) becomes (^(0))J (fiW - '(0,0,0) = t for j*i (=Fi,CiMlCU2/ci|CI) and the eigenvectors & {« = I,,.. m) corresponding to the trivial eigenvalue 0 become (&}J = '(0,0,0) for j / i , f {&)' = (0,C2,-CO- Since ^ ( 0 ) , ^ (t = l , . 9) is hyperbolic near w = 0. 11) has a lifespan Ts which is at least of order e - 3 for any ( £ R 2 . 19) tu=0 for i = 1 , . . , m a n d C £ R 2 Set P(A) =det(py(A) : i,j = l , . . , m ) . 20) (t = l t . . 21) = 0 dwi m = 0 for all ij,a.

Yu, Boundary value problem for quasi-linear hyperbolic system,, Duke Univ. Math. Ser. 5, 1985. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. M a t h . Soc. 275 (1983). Majda. The existenee of multi-dimensional shock fronts, Mem. Amer. M a t h . Soc. 281 (1983). Majda, One perspective on open problems in multi-dimensional conservation lav/s, IMA (1990), 217-237. Thomann, Mutti-dimensional shock fronss for second order wave equations, Comm. E. 12 (1987), 777-828. , Trans. Amer.

19) is equivalent to d21„-rS a „ dwlndw\ , . 26) u>=0 fori = 1 , . . 27) J=I for all ( £ R 2 Consequently we have proved the following Proposition 2. 8) follows from Proposition 2 . 1 . 50 3 Notations. To begin with, we introduce some notations that are used throughout the paper. Partial derivatives are denoted by dt dx\ ox2 We also use the angular derivative: SI — x\di — x-idi. = (AU A3,A3) is a multi-index. Let u = ^ ( u 1 . . 1) for 0 < 7 < 1/2. Then we define, for a non-negative integer fc, \du(t,x)\ = £^|30uU*)l m \du(t)\k = Y.