# On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations

### Jose Ernie C. Lope

University of the Philippines, Quezon City, Philippines### Marian P. Roque

University of the Philippines, Quezon City, Philippines### Hidetoshi Tahara

Sophia University, Tokyo, Japan

## Abstract

We study the singular nonlinear equation $tu_{t}=F(t,x,u,u_{x})$, where the function $F$ is assumed to be continuous in $t$ and holomorphic in the other variables. Under some growth conditions on the coefficients of the partial Taylor expansion of $F$, we show that if $F(t,x,0,0)$ is of order $O(\mu(t)^{\alpha})$ for some $\alpha\in[0,1]$ as $t\rightarrow0$ uniformly in some neighborhood of $x=0$, then the equation has a unique solution $u(t,x)$ with the same growth order.

## Cite this article

Jose Ernie C. Lope, Marian P. Roque, Hidetoshi Tahara, On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations. Z. Anal. Anwend. 31 (2012), no. 3, pp. 291–305

DOI 10.4171/ZAA/1461