Abstract
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 language | English |
|---|---|
| Pages (from-to) | 131-145 |
| Number of pages | 15 |
| Journal | Tokyo Journal of Mathematics |
| Volume | 36 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2013 Jun |
Keywords
- Formal solutions
- Genuine solutions
- Gevrey class
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Remarks on formal solution and genuine solutions for some nonlinear partial differential equations. / Yamazawa, Hiroshi.
In: Tokyo Journal of Mathematics, Vol. 36, No. 1, 06.2013, p. 131-145.Research output: Contribution to journal › Article
}
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
UR - http://www.scopus.com/inward/record.url?scp=84890158524&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84890158524&partnerID=8YFLogxK
U2 - 10.3836/tjm/1374497515
DO - 10.3836/tjm/1374497515
M3 - Article
VL - 36
SP - 131
EP - 145
JO - Tokyo Journal of Mathematics
JF - Tokyo Journal of Mathematics
SN - 0387-3870
IS - 1
ER -