On the Unique Solvability of Nonlinear Fuchsian Partial Differential Equations

Abstract

We consider a singular nonlinear partial differential equation of the form $$ (t\partial_t)^mu= F \Bigl( t,x,\bigl\{(t\partial_t)^j \partial_x^{\alpha}u \bigr\}_{(j,\alpha) \in I_m} \Bigr) $$ with arbitrary order $m$ and $I_m=\{(j,\alpha) \in \mathbb{N} \times \mathbb{N}^n \,;\, j+|\alpha| \leq m, j<m \}$ under the condition that $F(t,x,\{z_{j,\alpha} \}_{(j,\alpha) \in I_m})$ is continuous in $t$ and holomorphic in the other variables, and it satisfies $F(0,x,0) \equiv 0$ and $(\partial F/\partial z_{j,\alpha})(0,x,0) \equiv 0$ for any $(j,\alpha) \in I_m \cap \{|\alpha|>0 \}$. In this case, the equation is said to be a nonlinear Fuchsian partial differential equation. We show that if $F(t,x,0)$ vanishes at a certain order as $t$ tends to $0$ then the equation has a unique solution with the same decay order.