Remarks on formal solution and genuine solutions for some nonlinear partial differential equations

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Quchi ([2], [3]) found a formal solution (t, x) = SumkG0 uk(x)tk with Equation presented for some class of nonlinear partial differential equations. For these equations he showed that there exists a genuine solution u S(t,x) on a sector S with asymptotic expansion u S(t, x) -(t, x) as t → 0 in the sector S. These equations have polynomial type nonlinear terms. In this paper we study a similar class of equations with the following nonlinear terms Equation presented It is main purpose to get a solvability of the equation in a category u S(t, x) -0 as t → 0 in a sector S. We give a proof by the method that is a little different from that in [3], Further we give a remark that the similar class of equations has a genuine solution u S(t, x) with u S(t, x) -(t, x) as t → 0 in the sector S.

Original languageEnglish
Pages (from-to)131-145
Number of pages15
JournalTokyo Journal of Mathematics
Issue number1
Publication statusPublished - 2013 Jun



  • Formal solutions
  • Genuine solutions
  • Gevrey class

ASJC Scopus subject areas

  • Mathematics(all)

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