  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.
