Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 737 - 765, Volume 26, Issue 2, February 2010      doi:10.3934/dcds.2010.26.737

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Junde Wu - Department of Mathematics, Suzhou University, Suzhou, Jiangsu 215006, China (email)
Shangbin Cui - Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China (email)

Abstract: 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:  Free boundary problem, tumor growth, asymptotic stability, stable/unstable manifold, invariance structure.
Mathematics Subject Classification:  Primary: 34G20, 35R35, 35B35; Secondary: 76D27.

Received: December 2008;      Revised: May 2009;      Available Online: October 2009.