Suppose I have a smooth vector field that has the form $$ X(y) = \sum_j \lambda_j y^j \partial_j + \text{higher order terms}$$ for $\lambda_j>0$. Let $\Phi_t$ be the flow of $X$. Then it follows that $\Phi_t(y) \longrightarrow 0$ for $y$ near $0$ as $t \longrightarrow - \infty$.

I am now looking estimates on the $y$-derivatives. Precisely, suppose that $K$ is a compact neighborhood of $0$ that lies in the unstable manifold near the point $0$. I would like to have a statement like "For every multiindex $\alpha$, there exists a constant $C>0$ such that $$ \sup_{y \in K} |D^\alpha_y \Phi_t(y)| \leq C e^{t\lambda}$$ for all $t<0$ and $y \in K$, where $\lambda$ is the smallest eigenvalue of the linearization of $X$ at $0$"

Is some statement like this true? Where to find it or how do I prove it?