Uniqueness of the solution of nonlinear singular partial differential equations (Q1363223)

The existence and the uniqueness of the solution of nonlinear singular partial differential equations of the form \[ \Biggl(t{\partial\over\partial t}\Biggr)^mu= F\Biggl(t,x,\Biggl\{\Biggl(t{\partial\over\partial t}\Biggr)^j \Biggl({\partial\over\partial x}\Biggr)^\alpha u\Biggr\}_{j+|\alpha|\leq m,j<m}\Biggr)\tag{E} \] were discussed in [\textit{R. Gérard} and \textit{H. Tahara}, Publ. Res. Inst. Math. Sci., Kyoto Univ., 29, No. 1, 121-151 (1993; Zbl 0805.35015)]; though, the uniqueness in [loc. cit.] can be applied only to the solution with \[ \Biggl(t{\partial\over\partial t}\Biggr)^j u(t,x)= O(t^s)\quad(\text{as }t\to 0\text{ uniformly in }x), \] for \(j=0,1,\dots,m- 1\), for some \(s>0\). In this paper, the author proves the uniqueness of the solution of (E) under the following weaker assumption: \[ \Biggl(t{\partial\over\partial t}\Biggr)^j u(t,x)= O\Biggl({1\over (-\log t)^s}\Biggr)\quad (\text{as }t\to 0\text{ uniformly in }x), \] for \(j=0,1,\dots, m-1\), for some \(s>0\).