The asymptotic behavior of singular solutions of some nonlinear partial differential equations in the complex domain
Let \(u(t, x)\) \(((t, x)\in C\times C^d)\) be a solution of nonlinear partial differential equation \[ Lu= L(t,x,(t\partial_t)^{\alpha_0}u(t, x); (\alpha_0, \alpha')\in \mathbb{N}\times \mathbb{N}^d,|\alpha|\leq m)= 0 \] in a neighborhood of the origin, which is not necessary holomorphic on \(\{t=0\}\). Then under some conditions on \(L\) there exist \(u_n(t, x)\) represented by Mellin type integral and y > 0, vo such that \[ |u_n(,x)|\leq C_0 C^m\Gamma\Biggl({p_n\over\gamma}+ 1\Biggr)|t|^{p_n+ \nu_0}, \] \[ \Biggl| u(t,x)- \sum^{N-1}_{n=0} u_n(t, x)\Biggr|\leq AB^N\Gamma\Biggl({p_N\over\gamma}+ 1\Biggr)|t|^{p_n+ \nu_0}, \] where \(0= p_0< p_1<\cdots< p_n<\to +\infty\).
scientific article; zbMATH DE number 5543220