Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

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Abstract

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a 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 is sufficiently small, whereas if γ<γ* then this stationary solution is unstable.

MSC

34G20
35B35
35R35
47H20
76D27

Keywords

Free boundary problem
Tumor growth
Asymptotic stability
Center manifold
Local Lie group

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