ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35 Matsumoto, Waichiro, Ryukoku University/Université de Paris VI On the Cauchy-Kowalevskaya theorem of Nagumo type for systems We consider the Cauchy problem for system:

(C) \{ \begin{matrix}D_{t}u − ∑_{|α|≤m} A_{α}(t, x)D_{x}^{α}u = f(t, x), \\ u|_{t=t_{o}} = u_{o}(x), \end{matrix}

where m and N are natural numbers, u, u_{o}, and f are vectors of dimension N and A_{α} are square matrices of order N. When all coefficients are real analytic on t and x, H. Yamahara and I gave the necessary and sufficient condition in order that (C) has an unique solution for each real analytic u_{o} and f. (See [1].) On the other hand, when N = 1, the regulariry on t of the coefficients is relaxed to C^{0}-class. We consider the case of N > 1 and give an example, where (C) has not in general a classical solution for real analytic u_{o} and f in x if (m − 1)N > 4p. (The coefficients depend only on t and belong to C^{p−1,1}.) We also consider a sufficient condition, which is almost necessary in some restricted case. Ref. [1] W. Matsumoto and H. Yamahara, On the Cauchy-Kowalevskaya theorem for systems, Proc. Japan Acad. 67 Ser. A, No.6 (1991) 181–185.