  We consider the asymptotic limit of a diffuse interface model for
tumor-growth when a parameter $\varepsilon$ proportional to the thickness of
the diffuse interface goes to zero. An approximate solution which shows
explicitly the behavior of the true solution for small $\varepsilon$ will be
constructed by using matched expansion method. Based on the energy method, a
spectral condition in particular, we establish a smallness estimate of the
difference between the approximate solution and the true solution.

Analysis of PDEs