Special Session 26: Qualitative Analysis of Parabolic Equations

Of concern is a moving boundary problem modelling the growth of multicellular spheroids or in vitro tumors. This model consists of two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure in the tumor’s body, respectively. The driving mechanism of the evolution is governed by Darcy’s law. Finally surface tension effects on the moving interface counteract the internal pressure. Based on a centre manifold analysis, we prove that if the initial domain is sufficiently close to a Euclidean ball in a suitable Hoelder norm, then the solution exists globally and the corresponding domains converge exponentially fast to some (possibly translated) ball, provided the surface tension coefficient is larger than a positive threshold value γ∗. If the surface tension coefficient is less than γ∗ then the radially symmetric equilibrium centred at the origin is unstable.