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Linear Kovalevskian systems with time-dependent coefficients. (English) Zbl 0559.35011

For the linear Kovalevskian system \(U_ t=\sum^{n}_{h=1}A_ h(t)U_ x+B(t)U\) on \(R^ n\times [0,T]\) with \(U(x,0)=g(x)\), and \(A_ h(t)\), B(t) complex \(N\times N\) matrices in \(L^ 1[0,T]\), the Cauchy- Kovalevski theorem states that if g is holomorphic in the strip of \(C^ n:\) \(\{\) \(| Im z| <k\}\), then the problem has a unique solution analytic in x and absolutely continuous in t, for any t with \(\sup_{1\leq h\leq n}\{\| A_ h(s)\| ds\}\leq k\). This last inequality is an estimate of the time of existence of the solution. Several examples show that the estimate is too crude and motivate refined estimates which are derived in this paper.
Reviewer: E.Barron

MSC:

35F05 Linear first-order PDEs
35A10 Cauchy-Kovalevskaya theorems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] Bony J, Lecture Notes in Hath pp 82–
[2] Grothendieck A., Meraoires Amer.Math, Soc 16 (1955)
[3] HĂ–rmander, L. 1963. ”Linear Partial Differential Ope-rators”. Berlin: Springer-Verlag.
[4] Jannelli E., J.of Math.of Kyoto Un 21 (4) pp 715– (1981)
[5] Jannelli E., Comm.in P.D.E 7 (5) pp 537– (1982) · Zbl 0505.35051
[6] Martineau, A. 1963. ”Sur les fonctionnelles analytiques et la transformation de Fourier-Borel”. Vol. 11, 1–164. J.Analyse Math. · Zbl 0124.31804
[7] Mizohata S., Comm.Pure Appl.Math 14 pp 547– (1961) · Zbl 0105.07203
[8] Mizohata, S. 1973. ”The theory of partial differential equations”. Cambridge: University Press. · Zbl 0263.35001
[9] Ovciannikov L.V, Soviet Math. Dokl 6 pp 1025– (1965)
[10] Treves P., Academic Press (1975)
[11] Yamanaka T., Comment.Math.Univ.St.Paul 9 pp 7– (1960)
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