Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R ² of the form 0 < y < ρ(x, t), where ρ(x, t) is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little Hölder spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient γ is larger than a threshold value γ* > 0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.