TY - JOUR

T1 - Remarks on formal solution and genuine solutions for some nonlinear partial differential equations

AU - Yamazawa, Hiroshi

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - Formal solutions

KW - Genuine solutions

KW - Gevrey class

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U2 - 10.3836/tjm/1374497515

DO - 10.3836/tjm/1374497515

M3 - Article

AN - SCOPUS:84890158524

VL - 36

SP - 131

EP - 145

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -