TY - JOUR
T1 - Remarks on formal solution and genuine solutions for some nonlinear partial differential equations
AU - Yamazawa, Hiroshi
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 -