2007/12/15 by Cui, Shangbin · 1 citation
#34G20 #35R35 #47H20 #76D27 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.0712.2483
In this paper we study asymptotic behavior of solutions for a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. We first establish a general result for differential equations in Banach spaces possessing a local Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^∗ then the unique stationary solution is asymptotically stable modulo translations, provided the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small, whereas if γ< γ^∗ then this stationary solution is unstable.