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.

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)=+∞.

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。

研究两个分支的Degasperis-Procesi系统Cauchy问题当初值(u0,ρ0)在空间(H1(R)∩W1,∞(R))×(L2(R)∩L∞(R))时解的弱适定性问题.首先,利用特征线方法,把两个分支的Degasperis-Procesi系统化成一个ODE系统.其次,应用经典的常微分方程解的适定性理论,得到ODE系统解的存在唯一性.最后证明了两个分支的Degasperis-Procesi系统解的存在唯一性并给出解对初值的弱连续依赖性结论.

A nonlinear size-structured population model with distributed delay in birth process is stud-ied.The time lag in birth process and the internal competition are considered in the model.Sufficient conditions for the existence of steady states are obtained by using the method of semigroups.

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.

研究Degasperis-Procesi方程Cauchy问题当初值u0∈H1(R)∩W1,∞(R)时解的弱适定性.首先运用特征线将Degasperis-Procesi方程的Cauchy问题转化成一个ODE系统,其次利用ODE理论证明该ODE系统的解存在唯一,最后利用该ODE系统与原方程的关系,研究原方程解的存在唯一性,并给出原方程的解对初值的弱连续依赖性.

A mathematical model of hyperbaric oxygen therapy for chronic wound healing is studied.This model contains a nonlinear second-order parabolic equation describing the concentration of oxygen diffusion,a nonlinear first-order hyperbolic equation for capillary tip density and an ordinary differential equation for blood vessel density.The existence and uniqueness of the global solution is proved by applying the characteristic theory of hyperbolic equation,the Lp-theory,H(o)lder-estimate theory and Banach fixed point theorem.

In order to avoid the staircasing effect and edge blurring problem.A variable TVp norm was defined,and an adaptive TVp (ATVp)regularization model was proposed.Combining the AOS numeri-cal method,a complete ATVp regularization algorithm was shown,where p can be adaptive selected ac-cording to different image areas.The characteristics make the new model preserve the edge information better and avoid the staircasing effect while image restoration.Experiments showed that compared with the existing regularization models,it improved the restoration results in both visual effects and SBR and PSNR.

A mathematical model of immune cells inhibiting tumor immune evasion is studied.The mod-el involves strongly coupled parabolic PDEs.Applying the Lp -theory,Schauder-estimate and Banach Fixed Point Theorem,it is proven that the problem has an unique local solution.Then by extension meth-od,that the local solution is global is proven.

The model of image zooming based on fluid dynamics equation including convection term and diffusive term,have two ways of interpolation.The experimental results show the effective performance of the proposed model in image zooming.

A size-structured two-phase population model with delayed birth process is studied.The well-posedness for this model is established.It is shown that the solution of this model has balanced exponen-tial growth by means of semigroups.

The existence of the global-in-time entropy weak solutions is obtained for the Cauchy problem of the weakly dissipative Degasperis-Procesi equation with the initial value in L2 (R)∩L4 (R).

This paper is devoted to the study of the well-posedness of semilinear heat equation with initial data in Besov space of negative regular index by using the endpoint Strichartz estimates in Time-Space Besov space. More precisely, we prove that for initial data in B - 2 α+2 ,2 nα(α+1) 2 and B̄ - 2 α+2 ,2 nα(α+1) 2 , the semilinear heat equation has a unique local and global solution and the solutions are continuous to initial data with respect to time variable t under certain conditions of α and n. We also establish similar local results in subcritical cases for initial data in B - 2 α+2 ,r p under certain conditions of α, p, r and n. Copyright © 2013 Watam Press.

The growth and development of solid tumors occurs in two distinct phases :the avascular and the vascular phase.During the former growth phase the tumor remains in a diffusion-limited dormant state of a few millimeters in diameter, while during the later phase, invasion and metastasis do take place .A mathematical model of cancer cell breakout and invasion of normal tissue or extracellular matrix is stud -ied.The model consists of a system of four Reaction-diffusion-taxis partial differential equations and a de-generate parabolic partial differential equations .By using the parabolic L p-theory, the parabolic Schauder estimates, principle of comparison and the Banach fixed point theorem , it is proved that this system has a unique global solution.

Image inpainting is the process of filling in missing parts of damaged images based on information gleaned from surrounding areas.A new algorithm for inpainting based on the Navier-Stokes equation,which allows for inpainting inside of the inpainting region and denoising outside of the inpainting region,is outlined.The experimental results show the effective performance of the proposed model in restoring scratched photos,missing parts and even removal of entire objects from images.

A mathematical model of the response of spatially structured tumors to chemotherapy: drug kinetics is studied.The model is a free boundary problem of a partial differential equation.The tumor is assumed to comprise a single cell population which reproduces and dies at a rate dependent on the local drug concentration.By using the method of the Lp-theory for parabolic equations,the Banach fixed point theorem and the extensions method under some general conditions,it is proved that this problem has a unique global solution.And then,by applying upper and lower solution method in the theory of reaction diffusion equations under some other general conditions to obtain the stationary solution.It is proved that if 0w≤w*,there is no stationary solution;If w* w,there is a unique stationary solution(ws(r),Rs)(Rs0).

In this paper, a mathematical model for tumor growth with time delay in proliferation under indirect effect of inhibitor is studied. The delay represents the time taken for cells to undergo mitosis. Nonnegativity of solutions is investigated. The steady-state analysis is presented with respect to the magnitude of the delay. Existence of Hopf bifurcation is proved for some parameter values. Local and global stability of the stationary solutions are proved for other ones. The analysis of the effect of inhibitor's parameters on tumor's growth is presented. The results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding non-retarded problems for some parameter values.

For the inverse source problem where source term is f(t)(x)with f(t) unknown,the solution method similar to the first kind Volterra integral equation is obtained.And stability estimate and numerical examples are given.