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 solutions to mathematical models of tumor growth, equilibrium configurations within such models have remained largely unexplored. This was due, in part, to the complexity of the relevant free boundary problems, which is enhanced when the process deviates from radial symmetry. In this paper, we present the results of our investigation on the existence of nonspherical dormant states for a model of non-necrotic vascularized tumors. For the sake of clarity we perform the analysis on two-dimensional geometries, though our techniques are evidently applicable to the full three-dimensional problem. We rigorously show that there is, indeed, an abundance of steady states that are not radially symmetric. More precisely, we prove that at any radially symmetric stationary state with free boundary r = R-0 (which we first; show to exist), there begin infinitely many branches of equilibria that bifurcate from and break the symmetry of that radial state. The free boundaries along the bifurcation branches are of the form r = R-0 + epsilon cos(l theta) + Sigma (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 configuration.