CPAA
Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators
Liang Zhang X. H. Tang Yi Chen
Communications on Pure & Applied Analysis 2017, 16(3): 823-842 doi: 10.3934/cpaa.2017039
In this paper, we consider the following perturbed nonlocal elliptic equation
$\left\{ {\begin{array}{*{20}{l}}{ - {{\cal L}_K}u = \lambda u + f(x,u) + g(x,u),\;\;x \in \Omega ,}\\{u = 0,\;\;x \in \mathbb{R}{^N} \setminus \Omega ,}\end{array}} \right. $
where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.
keywords: Broken symmetry nonlocal elliptic equation variational methods
JIMO
Resilience analysis for project scheduling with renewable resource constraint and uncertain activity durations
Jian Xiong Yingwu Chen Zhongbao Zhou
Journal of Industrial & Management Optimization 2016, 12(2): 719-737 doi: 10.3934/jimo.2016.12.719
In the real world, project construction usually suffers various uncertain factors. Traditionally, uncertainty is modelled as either random variables or dynamic events. This paper addresses the case that the information about uncertainty is partially known when developing project schedules. The concept of resilience is introduced into project scheduling problems with resource constraint to measure a schedule's ability to absorb possible perturbation. The definition of resilience is given based on project equilibriums. Since the calculation of resilience is time intensive, a new surrogate measure is proposed to indicate schedule resilience. Correlation analysis between resilience and proposed surrogate measure is carried out. The experimental results suggest that the proposed surrogate measure is more appropriate to indicate resilience than makespan or total free slack.
keywords: Resource-constrained project scheduling surrogate measure. resilience analysis uncertain activity duration
DCDS-S
The bifurcations of solitary and kink waves described by the Gardner equation
Yiren Chen Zhengrong Liu
Discrete & Continuous Dynamical Systems - S 2016, 9(6): 1629-1645 doi: 10.3934/dcdss.2016067
In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.
keywords: kink waves. solitary waves The Gardner equation bifurcations phase analysis
DCDS
Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations
Yingshan Chen Shijin Ding Wenjun Wang
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5287-5307 doi: 10.3934/dcds.2016032
This paper is concerned with the Cauchy problem of the compressible Navier-Stokes-Smoluchowski equations in $\mathbb{R}^3$. Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence theory of global solutions in $H^3$ to the stationary profile is established. Moreover, when the initial perturbation is bounded in $L^p$-norm with $1\leq p< \frac{6}{5}$, we obtain the optimal convergence rates of the solution in $L^q$-norm with $2\leq q\leq 6$ and its first order derivative in $L^2$-norm.
keywords: the Cauchy problem global existence uniqueness Compressible Navier-Stokes-Smoluchowski equations optimal convergence rate.
PROC
On the uniqueness of singular solutions for a Hardy-Sobolev equation
Jann-Long Chern Yong-Li Tang Chuan-Jen Chyan Yi-Jung Chen
Conference Publications 2013, 2013(special): 123-128 doi: 10.3934/proc.2013.2013.123
In this paper, we consider the positive singular solutions for the following Hardy-Sobolev equation
                        $\Delta u+u^p+\frac{u^{2^*(s)-1}}{|x|^s}=0 $      in    $B_1 \setminus \left \{ 0 \right \},$
where $p>1, 0 < s < 2, 2^*(s)=\frac{2(n-s)}{n-2}$, $n\geq 3$ and $B_1$ is the unit ball in $ R^n$ centered at the origin. We prove that if $p>\frac{n+2}{n-2}$ then such solution is unique.
keywords: Hardy-Sobolev equation singular solution uniqueness of solutions.
DCDS-B
Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients
Shihe Xu Yinhui Chen Meng Bai
Discrete & Continuous Dynamical Systems - B 2016, 21(3): 997-1008 doi: 10.3934/dcdsb.2016.21.997
In this paper we study a free boundary problem for the growth of avascular tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the supply of external nutrients is periodic. We mainly study the long time behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a positive periodic state.
keywords: global solution periodic solution free boundary problem asymptotic behavior. Tumors
JIMO
Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems
Mei Ju Luo Yi Zeng Chen
Journal of Industrial & Management Optimization 2016, 12(1): 1-15 doi: 10.3934/jimo.2016.12.1
In this paper, we consider the class of stochastic generalized Nash equilibrium problems (SGNEP). Such problems have a wide range of applications and have attracted significant attention recently. First, using the first order optimality condition of SGNEP and the nonlinear complementary function, we present an expected residual minimization (ERM) model for the case when the involved functions are not continuously differentiable. Then, we introduce a smoothing function, depending on a smoothing parameter, to yield a smooth approximate ERM model. We further show that the solutions of this smooth ERM model converge to the solutions of the original ERM model as the smoothing parameter tends to zero. Since the ERM formulation contains an expectation, we further propose a sample average approximate problem for the ERM model. Moreover, we show that the global optimal solutions of these approximate problems converge to the global optimal solutions of the ERM problem with probability one. Here, convergence can be achieved in two ways. One is to fix the smoothing parameter, the other is to let the smoothing parameter tend to zero as the sample increases.
keywords: sample average approximation Stochastic generalized Nash equilibrium smooth approximation expected residual minimization exponential convergence.
DCDS-B
Bernoulli shift for second order recurrence relations near the anti-integrable limit
Yi-Chiuan Chen
Discrete & Continuous Dynamical Systems - B 2005, 5(3): 587-598 doi: 10.3934/dcdsb.2005.5.587
We extend the anti-integrability theory of Aubry to non-autonomous twist maps between symplectic spaces to show the shift dynamics can be embedded in a natural way. Examples are given to illustrate that the embedded shift can be a full shift, a subshift of finite type or of infinite type.
keywords: Anti-integrable limit Symbolic dynamics. Non-autonomous maps
DCDS-B
On global boundedness of the Chen system
Fuchen Zhang Xiaofeng Liao Chunlai Mu Guangyun Zhang Yi-An Chen
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1673-1681 doi: 10.3934/dcdsb.2017080

This paper deals with the open problem of the global boundedness of the Chen system based on Lyapunov stability theory, which was proposed by Qin and Chen (2007). The innovation of the paper is that this paper not only proves the Chen system is global bounded for a certain range of the parameters according to stability theory of dynamical systems but also gives a family of mathematical expressions of global exponential attractive sets for the Chen system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained.

keywords: Chen system Lyapunov stability attractor global exponential attractive sets
KRM
A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions
Yingshan Chen Mei Zhang
Kinetic & Related Models 2016, 9(3): 429-441 doi: 10.3934/krm.2016001
In this paper, we establish a new blowup criterion for the strong solutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In [13], Wen, Yao, and Zhu prove that if the strong solutions blow up at finite time $T^*$, the mass in $L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extend Wen, Yao, and Zhu's work in the sense of the concentration of mass in $BMO(\Omega)$ norm at $T^*$. The method can be applied to study the blow-up criterion in terms of the concentration of density in $BMO(\Omega)$ norm for the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a byproduct, we can also improves the corresponding result about Navier-Stokes equations in [11]. Moreover, the appearance of vacuum is allowed in the paper.
keywords: strong solution Liquid-gas two-phase flow model BMO criterion vacuum.

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