Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations (Q1020617)

The authors study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The velocity of the fluid fulfills Stokes equations with a source term determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. They prove first that this problem has a unique radially symmetric stationary solution. By using the so-called Hanzawa transformation and various tools from functional analysis the authors are able to prove the following: There exists a threshold value \(\gamma_{*} > 0\) for the surface tension coefficient \(\gamma\), such that in the case \(\gamma > \gamma_{*}\) this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.