## Journal of the Mathematical Society of Japan

### Uniqueness of the solution of nonlinear totally characteristic partial differential equations

Hidetoshi TAHARA

#### Abstract

Let us consider the following nonlinear singular partial differential equation $(t \partial/\partial t)^m u = F ( t,x, \{(t \partial/\partial t)^j (\partial/\partial x)^{\alpha} u \}_{j+\alpha \leq m, j in the complex domain with two independent variables $(t,x) \in \mathbf{C}^2$. When the equation is of totally characteristic type, this equation was solved in [2] and [9] under certain Poincaré condition. In this paper, the author will prove the uniqueness of the solution under the assumption that $u(t,x)$ is holomorphic in $\{(t,x) \in \mathbf{C}^2; 0 < |t| for some $r>0$, $\theta >0$, $R>0$ and that it satisfies $u(t,x) = O(|t|^a )$ (as $t \longrightarrow 0$) uniformly in $x$ for some $a>0$. The result is applied to the problem of removable singularities of the solution.

#### Article information

Source
J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1045-1065.

Dates
First available in Project Euclid: 14 June 2006

https://projecteuclid.org/euclid.jmsj/1150287303

Digital Object Identifier
doi:10.2969/jmsj/1150287303

Mathematical Reviews number (MathSciNet)
MR2183583

Zentralblatt MATH identifier
1163.35302

#### Citation

TAHARA, Hidetoshi. Uniqueness of the solution of nonlinear totally characteristic partial differential equations. J. Math. Soc. Japan 57 (2005), no. 4, 1045--1065. doi:10.2969/jmsj/1150287303. https://projecteuclid.org/euclid.jmsj/1150287303

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