  We consider a system of $n$-th order nonlinear quasilinear partial
differential equations of the form
  $${\bf u}_t + \mathcal{P}(\partial_{\bf x}^{\bf j}){\bf u}+{\bf g} \left(
{\bf x}, t, \{\partial_{\bf x}^{{\bf j}} {\bf u}\}) =0; {\bf {u}}({\bf x}, 0) =
{\bf {u}}_I({\bf x})$$
  with $\mathbf{u}\in\CC^{r}$, for $ t\in (0,T)$ and large $|{\bf x}|$ in a
poly-sector $S$ in $\mathbb{C}^d$ ($\partial_{\bf x}^{\bf j} \equiv
\partial_{x_1}^{j_1} \partial_{x_2}^{j_2} ...\partial_{x_d}^{j_d}$ and
$j_1+...+j_d\le n$). The principal part of the constant coefficient $n$-th
order differential operator $\mathcal{P}$ is subject to a cone condition. The
nonlinearity ${\bf g}$ and the functions $\mb u_I$ and $\mb u$ satisfy
analyticity and decay assumptions in $S$.The paper shows existence and
uniqueness of the solution of this problem and finds its asymptotic behavior
for large $|\bf x|$. Under further regularity conditions on $\mb g$ and $\mb
u_I$ which ensure the existence of a formal asymptotic series solution for
large $|\mb x|$ to the problem, we prove its Borel summability (and
automatically its asymptoticity) to an actual solution $\mb u$.In special cases
motivated by applications we show how the method can be adapted to obtain
short-time existence, uniqueness and asymptotic behavior for small $t$,without
size restriction on the space variable.
