A parabolic-hyperbolic system modeling the growth of a tumor

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Li, Rui¹  Hu, Bei² 

[1] Renmin Univ China, Sch Math, Beijing 100872, Peoples R China

[2] Univ Notre Dame, Dept Appl Computat Math & Stat, Notre Dame, IN 46556 USA

关键词: Free boundary;    Shooting method;    Singular integro-differential equations;    Uniqueness;    Simulation;   

DOI  :  10.1016/j.jde.2019.01.020

来源: Elsevier

【 摘 要 】

In this paper, we consider a model with tumor microenvironment involving nutrient density, extracellular matrix and matrix degrading enzymes, which satisfy a coupled system of PDEs with a free boundary. For this coupled parabolic-hyperbolic free boundary problem, we prove that there is a unique radially symmetric solution globally in time. The stationary problem involves an ODE system which is transformed into a singular integro-differential equation. We establish a well-posed theorem for such general types of equations by the shooting method; the theorem is then applied to our problem for the existence of a stationary solution. In addition, for this highly nonlinear problem, we also prove the uniqueness of the stationary solution, which is a nontrivial result. Moreover, numerical simulations indicate that the stationary solution is likely locally asymptotically stable for reasonable range of parameters. (c) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free