TY - JOUR

T1 - Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth

AU - Cui, Shangbin

AU - Escher, Joachim

N1 - Funding information: This work on the part of the first author is supported by the National Natural Science Foundation of China under the grant number 10471157. The authors would like to thank the anonymous referee for valuable suggestions improving the first version of this paper.

PY - 2008/4/1

Y1 - 2008/4/1

N2 - 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 Cm+-norm with m3 and (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*. In the case 0* the radially symmetric equilibrium is unstable.

AB - 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 Cm+-norm with m3 and (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*. In the case 0* the radially symmetric equilibrium is unstable.

KW - Centre manifold

KW - Moving boundary problem

KW - Stability

KW - Surface tension

KW - Tumor growth

UR - http://www.scopus.com/inward/record.url?scp=41849134265&partnerID=8YFLogxK

U2 - 10.1080/03605300701743848

DO - 10.1080/03605300701743848

M3 - Article

AN - SCOPUS:41849134265

VL - 33

SP - 636

EP - 655

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 4

ER -