The problem is something like:. $$ B(u(x_1,x_2,...,x_n))+g(x_1,...,x_n,u)=D(u)(A(x)), $$ and we have the initial conditions $u(0)=D(u)(0)=0$.
$u=(u_1,...,u_n)$ is the unknown vector field, and it is a vector field from the $n$-dimensional euclidean space to the $n$-dimensional euclidean space; $g$ is a low degree polynomial vector field from the $2n$-dimensional euclidean space to the n-dimensional one, $x=(x_1,...,x_n)$ is the vector of variables; $D(u)$ is the derivative of $u$ (the Jacobian matrix); $0$ is the zero vector; $A$ is a matrix such that all its eigenvalues are positive, while $B$ is matrix such that all its eigenvalues are negative.
The question is: Is it possible to prove that there exists a neighborhood of $0$, say $U$, such that the problem has a single solution $u$ over $U$, and the function $\lVert u\rVert$ is a slow growing function, where $\lVert u\rVert$ is the euclidean norm of $u$?.