Analysis of a solid avascular tumor growth model with time delays in proliferation process

In this paper we study a free boundary problem modeling solid avascular tumor growth. The model is based on the reaction–diffusion dynamics and mass conservation law. The model is considered with time delays in proliferation process. The quasi-steady-state (i.e., d=0) is studied by Foryś and Bodnar [see U. Foryś, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling 37 (2003) 1201–1209]. In this paper we mainly consider the case d>0. In the case considered by Foryś and Bodnar, the model is reduced to an ordinary differential equation with time delay, but in the case d>0 the model cannot be reduced to an ordinary differential equation with time delay. By Lp theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the long time asymptotic behavior of the solutions under some conditions.