Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane

Abstract

We prove existence and uniqueness results for nonlinear third order partial differential equations of the form $$ f_t - f_{yyy} = \sum_{j=0}^3 b_j (y, t; f) ~f^{(j)} + r(y, t) $$ where superscript $j$ denotes the $j$-th partial derivative with respect to $y$. The inhomogeneous term $r$, the coefficients $b_j$ and the initial condition $f(y,0)$ are required to vanish algebraically for large $|y|$ in a wide enough sector in the complex $y$-plane. Using methods related to Borel summation, a unique solution is shown to exist that is analytic in $y$ for all large $|y|$ in a sector. Three partial differential equations arising in the context of Hele-Shaw fingering and dendritic crystal growth are shown to be of this form after appropriate transformation, and then precise results are obtained for them. The implications of the rigorous analysis on some similarity solutions, formerly hypothesized in two of these examples, are examined.