Stationary Solutions of a Model for the Growth of Tumors and a Connection between the Nonnecrotic and Necrotic Phases

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

“…Global well-posedness was also established in [4] [5]. The existence of necrotic stationary solutions for nonlinear and strictly increasing functions f and g was established in [2] [17]. Similarly, in this paper we always assume (A1) f ∈ C 1 [0, +∞) is strictly increasing and f is bounded on [0, +∞), and f (0) = 0; (A2) g ∈ C 1 [0, +∞) is strictly increasing and g(σ) = 0 for someσ ∈ (σ, +∞); (A3) g(σ) + ν ≥ 0.…”

Initial/boundary-value problems of tumor growth within a host tissue

“…Growth was described under the main assumption of constant cell density, by relating the volume variation of the tumor mass to birth and death of cells triggered by nutrient supply. In most cases simple in vitro geometries were considered, such as spheroids, and qualitative analyses of the resulting free boundary problems were detailed [12,13,20,21,25,26,27].…”

Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation

“…Most of those models are based on the reaction-diffusion equations and mass conservation law. Analysis of such mathematical models has drawn great interest, and many results have been established; compare [13][14][15][16][17][18][19][20][21][22] and references therein. Many numerical results have been performed; compare [1,2,13,23,24] and references therein.…”

“…For process of necrotic core formation, the study by using the method of mathematical models was initiated by Byrne in [1]. Recently this study has drawn the attention of some other researchers; compare Bueno et al [15], Cui [16], Foryś and Mokwa-Borkowska [13], and references cited therein. But all of them are related to quasistationary version; that is, = 0, and Cui [16] and Foryś and Mokwa-Borkowska [13] studied the particular cases in which ( ) = Γ and ( ) = , respectively, where Γ and are two constants.…”

“…The model we studied in this paper is established by modifying the model studied in [15] by introducing a time delay in proliferation as that in Byrne [1]. In their model, the cell proliferation rate in unit volume is ( ), where is a continuous function which satisfies some conditions; that is, (4) is as follows ( = 0):…”