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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 3, 2012, pp. 291–305
DOI: 10.4171/ZAA/1461

Published online: 2012-07-11

On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations

Jose Ernie C. Lope[1], Marian P. Roque[2] and Hidetoshi Tahara[3]

(1) University of the Philippines, Quezon City, Philippines
(2) University of the Philippines, Quezon City, Philippines
(3) Sophia University, Tokyo, Japan

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.

Keywords: Unique solvability, singular partial di

Lope Jose Ernie, Roque Marian, Tahara Hidetoshi: On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations. Z. Anal. Anwend. 31 (2012), 291-305. doi: 10.4171/ZAA/1461