DCDS-B
Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
Jian-Guo Liu Min Tang Li Wang Zhennan Zhou
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-25 doi: 10.3934/dcdsb.2018297

In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density $ n$ is governed by the Darcy law via the pressure $ p(n) = n^{γ}$. For finite $ γ$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $ γ \to ∞$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

keywords: Tumor growth model free boundary limit Hele-Shaw flow model
JIMO
A differential equation method for solving box constrained variational inequality problems
Li Wang Yang Li Liwei Zhang
Journal of Industrial & Management Optimization 2011, 7(1): 183-198 doi: 10.3934/jimo.2011.7.183
In this paper, we discuss a system of differential equations based on the projection operator for solving the box constrained variational inequality problems. The equilibrium solutions to the differential equation system are proved to be the solutions of the box constrained variational inequality problems. Two differential inclusion problems associated with the system of differential equations are introduced. It is proved that the equilibrium solution to the differential equation system is locally asymptotically stable by verifying the locally asymptotical stability of the equilibrium positions of the differential inclusion problems. An Euler discrete scheme with Armijo line search rule is introduced and its global convergence is demonstrated. The numerical experiments are reported to show that the Euler method is effective.
keywords: global convergence. differential inclusion asymptotical stability differential equations Box constrained variational inequality problem
DCDS
Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent
Yinbin Deng Shuangjie Peng Li Wang
Discrete & Continuous Dynamical Systems - A 2012, 32(3): 795-826 doi: 10.3934/dcds.2012.32.795
In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
keywords: variational methods. critical exponent Multiple solutions
DCDS
Infinitely many radial solutions to elliptic systems involving critical exponents
Yinbin Deng Shuangjie Peng Li Wang
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 461-475 doi: 10.3934/dcds.2014.34.461
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
keywords: elliptic systems Radial solution (PS) condition. critical exponent

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