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### Open Access Journals

CPAA

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.

JIMO

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.

DCDS-S

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.

DCDS

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.

PROC

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.

$\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.

DCDS-B

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

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.

DCDS-B

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.

DCDS-B

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.

KRM

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.

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