Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth

作者：Cui, Shangbin; Escher, Joachim^*

来源：Communications in Partial Differential Equations, 2008, 33(4): 636-655.

DOI：10.1080/03605300701743848

• centre manifold
• moving boundary problem
• stability
• surface
• tension
• tumor growth

We study a moving boundary problem modeling the growth of multicellular spheroids or in vitro tumors. This model consists of two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure in the tumor's body, respectively. The driving mechanism of the evolution of the tumor surface is governed by Darcy's law. Finally surface tension effects on the moving boundary are taken into account which are considered to counterbalance the internal pressure. To put our analysis on a solid basis, we first state a local well-posedness result for general initial data. However, the main purpose of our study is the investigation of the asymptotic behaviour of solutions as time goes to infinity. As a result of a centre manifold analysis, we prove that if the initial domain is sufficiently close to a Euclidean ball in the C(m mu)-norm with m >= 3 and mu is an element of (0,1), then the solution exists globally and the corresponding domains converge exponentially fast to some (possibly shifted) ball, provided the surface tension coefficient is larger than a positive threshold value gamma*. In the case 0<gamma<gamma* the radially symmetric equilibrium is unstable.

• 出版日期2008
• 单位中山大学