On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumors

Despite their great importance in determining the dynamic evolution of solu tions to mathematical models of tumor growth, equilibrium configurations wi thin such models have remained largely unexplored. This was due, in part, t o the complexity of the relevant free boundary problems, which is enhanced when the process deviates from radial symmetry. In this paper, we present t he results of our investigation on the existence of nonspherical dormant st ates for a model of non-necrotic vascularized tumors. For the sake of clari ty we perform the analysis on two-dimensional geometries, though our techni ques are evidently applicable to the full three-dimensional problem. We rig orously show that there is, indeed, an abundance of steady states that are not radially symmetric. More precisely, we prove that at any radially symme tric stationary state with free boundary r = R-0 (which we first; show to e xist), there begin infinitely many branches of equilibria that bifurcate fr om and break the symmetry of that radial state. The free boundaries along t he bifurcation branches are of the form r = R-0 + epsilon cos(l theta) + Si gma (infinity)(n=2) f(n)(theta)epsilon (n), where l = 2, 3,... and \epsilon \ < epsilon (0); each choice of e and a determines a non-radial steady conf iguration.