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April  2010, 26(2): 737-765. doi: 10.3934/dcds.2010.26.737

Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth

Junde Wu 1, and  Shangbin Cui 2,

1. 

Department of Mathematics, Suzhou University, Suzhou, Jiangsu 215006, China
2. 

Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275

Received  December 2008 Revised  May 2009 Published  October 2009

In this paper we first study local phase diagram of an abstract parabolic differential equation in a Banach space such that the equation possesses an invariance structure under a local Lie group action. Next we use this abstract result to study a free boundary problem modeling the growth of non-necrotic tumors in the presence of inhibitors. This problem contains two reaction-diffusion equations describing the diffusion of the nutrient and the inhibitor, respectively, and an elliptic equation describing the distribution of the internal pressure. There is also an equation for the surface tension to govern the movement of the free boundary. By first performing some reduction processes to write this free boundary problem into a parabolic differential equation in a Banach space, and next using a new center manifold theorem established recently by Cui [8], and the abstract result mentioned above, we prove that under suitable conditions the radial stationary solution is locally asymptotically stable under small non-radial perturbations, and when these conditions are not satisfied then such a stationary solution is unstable. In the second case we also give a description of local phase diagram of the equation in a neighborhood of the radial stationary solution and construct its stable and unstable manifolds. In particular, we prove that in the unstable case, if the transient solution exists globally and is contained in a neighborhood of the radial stationary solution, then the transient solution will finally converge to a nearby radial stationary solution uniquely determined by the initial data.
Keywords: stable/unstable manifold, asymptotic stability, invariance structure., tumor growth, Free boundary problem.
Mathematics Subject Classification: Primary: 34G20, 35R35, 35B35; Secondary: 76D2.
Citation: Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
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