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

Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord

Pages: 929 - 943, Volume 21, Issue 3, July 2008      doi:10.3934/dcds.2008.21.929

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Fujun Zhou - Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China (email)
Shangbin Cui - Institute of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China (email)

Abstract: In this paper we study a multidimensional moving boundary problem modeling the growth of tumor cord. This problem contains two coupled elliptic equations defined in a bounded domain in $R^2$ whose boundary consists of two disjoint closed curves, one fixed and the other moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation. By using the functional analysis method based on applications of the theory of analytic semigroups, we prove that (1) this problem is locally well-posed in Hölder spaces, (2) it has a unique radially symmetric stationary solution, and (3) this radially symmetric stationary solution is asymptotically stable for arbitrary sufficiently small perturbations in these Hölder spaces.

Keywords:  Moving boundary problem, tumor cord, well-posedness, stability.
Mathematics Subject Classification:  Primary: 35R35; 35B35 Secondary: 76D27.

Received: June 2007;      Revised: December 2007;      Available Online: April 2008.