Symmetry-breaking bifurcations for free boundary problems

Journal of Mathematical Analysis and Applications

Cui

2008

This paper is devoted to the study of the bifurcation of a free boundary problem modeling the growth of tumors with the effect of surface tension being considered. The existence of infinitely many branches of bifurcation solutions is proved. The method of analysis is based on reducing the problem to an operator equation in certain Hölder space with a nonlinear Fredholm operator of index 0. The desired result then follows from the Crandall-Rabinowitz bifurcation theorem.

Section: Introductionmentioning

confidence: 99%

Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors

Zhou

Journal of Mathematical Analysis and Applications

Cui

2008

“…The simpler model, as presented in [4,8,11], has been studied extensively by different authors, see, e.g. [3,6,7,9] and the references therein. In particular it is shown in these papers that if the parameters belong to an appropriate range, then the mathematical formulation possesses a unique radially symmetric solution.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Radially symmetric growth of nonnecrotic tumors

Nonlinear Differ. Equ. Appl.

Matioc

2009

Abstract. The growth of tumors is an important subject in recent research. We present here a mathematical model for the growth of nonnecrotic tumors in all the three regimes of vascularisation. This leads to a freeboundary problem which we treat by means ODE techniques. We prove the existence of a unique radially symmetric stationary solution. It is also shown that, if the initial tumor is radially symmetric, there exists a unique radially symmetric solution of the evolution equation, which exists for all times. The asymptotic behaviour of this solution will be discussed in relation to the parameters characterizing cell proliferation and cell death.

“…However the model presented here includes two moving boundaries, one parametrising the boundary of the necrotic core and one for the outer boundary of the tumor, both of them having infinitely many degrees of freedom. This fact makes the problem more involved in comparison to other models which either neglect the necrotic core [3,[6][7][8][9][10] or consider only the radially symmetric problem when the tumors are annular domains [10,17,18].…”

Section: The Mathematical Modelmentioning

confidence: 99%

Analysis of a Mathematical Model Describing Necrotic Tumor Growth

Modelling, Simulation and Software Concepts for Scientific-Technological Problems

Matioc

2011

Abstract. In this paper we study a model describing the growth of necrotic tumors in different regimes of vascularisation. The tumor consists of a necrotic core of death cells and a surrounding nonnecrotic shell. The corresponding mathematical formulation is a moving boundary problem where both boundaries delimiting the nonnecrotic shell are allowed to evolve in time. We determine all radially symmetric stationary solutions of the problem and reduce the moving boundary problem into a nonlinear evolution. Parabolic theory provides us the perfect context in order to show local well-posed of the problem for small initial data.