Analysis of a free-boundary vascularized tumor growth model with time delays and inhibitor action

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This paper investigates a free boundary problem focused on the growth dynamics of vascularized tumors, incorporating time delays and the impact of inhibitors. Unlike existing vascularized tumor models with inhibitors, our model contains time delays, which represent the intrinsic delays of cell proliferation. The substance exchange between the vascularized tumor and its surrounding vascular network is represented by the Robin boundary. The problem comprises a system of nonlinear reaction-diffusion equations that characterizes the nutrient concentration u(r, t) and the inhibitor concentration v(r, t), together with an ordinary differential equation representing the tumor radius R(t). Firstly, it is shown that the model possesses at least one steady-state solution under certain sufficient conditions. Next, we demonstrate that the quasi-steady state system possesses a non-negative solution and analyze its stability. Finally, we establish the existence and uniqueness of the global classical solution and further analyze the asymptotic stability of the steady-state solution. Our results demonstrate that time delays in cell proliferation do not change the overall evolution trend of the tumor, but only slow the tumor growth process.