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.

In this paper, we study a free boundary problem of tumor growth with necrotic core. The model is a parabolic-hyperbolic partial differential equations, which is composed of three first-order nonlinear hyperbolic equations, a parabolic equation and an ordinary differential equation. First, we obtained the approximation model by polishing the Heaviside function, and then proved the existence and uniqueness of the solution of the approximation model. In addition, we improved the regularity of solution of the approximate problem by using the characteristic curves method, and finally proved the global existence of the weak solution of the original problem by the convergence.

This paper investigates a free boundary tumor model with timedependent in the presence of inhibitors. The model consists of two diffusion equations representing nutrients and inhibitors respectively, and an ordinary differential equation describing the radius of the tumor R(t). We know that angiogenesis is not a steady-state process, in general, it changes over time, so it is reasonable to assume that tumors stimulate angiogenesis at a rate proportional to alpha(t). We find the properties of the tumor radius R(t) is greatly tied to the properties of alpha(t). When alpha (t) is time-dependent, we prove that for any sufficiently small c(1): If alpha(t) remains uniformly bounded, then R(t) also remains uniformly bounded; If alpha(t) tends to zero as t ->infinity, so does the tumor radius R(t); If lim(t ->infinity) inf alpha (t) > 0, then lim(t ->infinity) inf R (t) > 0. Moreover, the global asymptotic stability of the steady-state solution is proved, and it is surprising to find that when u +v over bar is sufficiently small and lambda/mu u over bar < c(1) <= c(2), the solution will blow up.

In this paper, we research the issue of the free boundary of vascularized tumor growth using a T-periodic supply ψ(t) of outside nutrients and inhibitors. The model consists of two reaction diffusion equations, an elliptic equation and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor concentrations. The internal pressure distribution is described by the elliptic equation. The ODE describes the boundary value condition of the tumor model. After some meticulous mathematical analysis of the model system, we prove the existence and uniqueness of the radially symmetric T-periodic positive solution with u˜≤min0≤t≤T⁡ψ(t), where u˜ is a parameter, denoting a threshold concentration for proliferation. Next, we further demonstrate the existence of a μ⁎>0 such that (u⁎(r,t),v⁎(r,t),p⁎(r,t),R⁎(t)) is linearly stable for μ<μ⁎ and linearly unstable for μ>μ⁎ under perturbations that are not radially symmetric, where μ is a constant, representing the "intensity" of mitosis-induced cell growth.

A mathematical model of checkpoint inhibitor targeted therapy for human melanoma is inves-tigated.The model consists of twelve coupled reaction-diffusion equations,which includes free bound-ary conditions and discontinuous terms.By transforming the free boundary problem into the fixed boundary problem,using the Lp theory of the parabolic equation and the Schauder fixed point theorem,and combining with the method of function approximation,the existence of the global weak solution of the mathematical model is obtained.

In this paper, we study the linear stability of the free boundary problem for tumors. The model is a coupled system of PDE with Robin boundary, which involves concentration of nutrients, concentration of inhibitor and pressure. The presence of inhibitor in this model affects the diffusion of nutrient. We establish the existence and uniqueness of the radially-symmetric solution (us,vs,ps,Rs). We further prove that there exists a threshold value μ⁎ such that (us,vs,ps,Rs) is linearly stable under non-radially symmetric perturbations for μ∈(0,μ⁎) and linearly unstable for μ>μ⁎.

In this paper, we study a time-delayed free boundary of tumor growth with Gibbs-Thomson relation in the presence of inhibitors. The model consists of two reaction diffusion equations and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor diffusion within tumors and take into account the Gibbs-Thomson relation at the outer boundary of the tumor. The tumor radius evolution is described by the ordinary differential equation. It is assumed that the regulatory apoptosis process takes longer than the natural apoptosis and proliferation processes. We first show the existence and uniqueness of the solution to the model. Next, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is a constant. Finally, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is bounded. The result implies that, under certain conditions, the tumor will probably become dormant or will finally disappear. The conclusions are illustrated by numerical computations.

In this paper, we study a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis. The characteristic of this model is that both vascularization and apoptosis regulation is considered. In mathematical form, this model is expressed as a free boundary problem with Robin boundary. We prove the existence and uniqueness of the global solution and their asymptotic behavior. The effects of vascularization parameters and apoptosis regulation parameters on tumor are discussed. Depending on the importance of regulating the apoptosis rate, the tumor will tend to the unique steady state or eventually disappear. For some parameter values, the final results show that the dynamic behavior of the solutions of our model is analogous to the quasi-stationary solutions. Our results are also verified by numerical simulation.

In this paper, we study the parabolic–hyperbolic system about the growth of a tumor. The model is a coupled system of PDEs with Robin boundary, which involves nutrient density, extracellular matrix and matrix degrading enzyme. By transforming the free boundary into a fixed boundary and using strict mathematical analysis, we can prove the existence and uniqueness of the radially symmetric stationary solution. By the fixed point theorem, we obtain the existence and uniqueness of the radially symmetric solution globally in time.

研究了一个具有Robin自由边界的双曲肿瘤生长数学模型,该模型包含了一个描述营养物浓度变化的椭圆型方程,一个描述肿瘤半径的常微分方程和描述肿瘤细胞生长的两个双曲型偏微分方程.本文通过特征线方法结合Banach不动点定理证明了该模型整体解的存在性和唯一性.最后证明当KR=0时,有(limt→∞)R(t)=∞.

In this paper, we study the periodic Hunter-Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.

研究了一个具有坏死核的双曲型肿瘤生长的Robin自由边界问题.该模型包含了一个描述营养物浓度变化的椭圆型方程,一个描述肿瘤半径的常微分方程和三个分别描述增殖细胞,休眠细胞和死亡细胞演化的一阶非线性双曲偏微分方程.通过特征线方法和Banach不动点定理证明了该模型整体解的存在唯一性.同时证明了当KR=0时,limt→+∞R(t)=+∞.

本文主要研究广义的Camassa-Holm方程Cauchy问题当初值u0在空间(R)∩W1,∞(R)时解的弱适定性.首先运用特征线把广义的Camassa-Holm方程转化成类似常微分方程(Ordinary Differential Equation,ODE)的方程.其次运用ODE理论证明新方程解的局部存在唯一性.最后利用新方程与原方程的关系,证明原方程解的局部存在唯一性并且给出解对初值的弱连续依赖性.

研究癌症疫苗和检查点抑制剂联合治疗的数学模型,该模型为包含九个相互耦合反应扩散方程的方程组。先通过运用Banach不动点定理、抛物型方程的L p 估计证明了模型的局部解的存在唯一性,然后利用延拓方法得到了整体解的存在唯一性。

The metabolic model of colon cancer cells is studied. The model contains five coupled reaction diffusion equations, in which some equations involve discontinuous terms. It is proved that this problem has a global solution by using the Lp-theory for parabolic equations, the Schauder Fixed Point Theorem and approximation method.

In this paper we study a granuloma model in visceral leishmaniasis, which contains eleven coupled reaction diffusion equations. The existence and uniqueness of the model in the local solution is proved by using the Banach Fixed Point Theorem and theory of parabolic equation. Then, the existence and uniqueness of the global solution are obtained by using the extension method.

This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa-Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.

研究一个具有第三边界坏死核的肿瘤生长的数学模型,该模型包含了一个抛物型方程和一个常微分方程.假设肿瘤的生长由营养物浓度决定,并且肿瘤形状为球对称.用严格的数学分析的方法,证明了该模型的稳态解的存在唯一性.

In this paper we consider a free boundary problem modeling tumor growth with angiogenesis. The model is a free boundary problem of a system of partial differential equations. With Robin boundary, the model contains an ordinary differential equation describing the radius of tumor cell and two parabolic equations describing the evolution of nutrient concentration and inhibitor concentration, respectively. We study the number of the stationary solution of above problem, the existence and uniqueness of local solution and global solution, and the asymptotic behavior of the solution.

作者以在教学中遇到的孤立奇点处留数的定义和调和函数的可微性为例,谈谈研究式学习的可行性,希望能对学生的学习提供有效的案例和有益启发.