## Journals

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### Open Access Journals

JIMO

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.

DCDS

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 $(\star)$.

DCDS

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.

DCDS-B

We study a mathematical model for the interaction of HIV infection
and CD4$^+$ T cells. Local and global analysis is carried out. Let
$N$ be the number of HIV virus produced per actively infected T
cell. After identifying a critical number $N_{crit}$, we show that
if $N\le N_{crit},$ then the uninfected steady state $P_{0}$ is the
only equilibrium in the feasible region, and $P_{0}$ is globally
asymptotically stable. Therefore, no HIV infection persists. If
$N>N_{crit},$ then the infected steady state $P^$* emerges as
the unique equilibrium in the interior of the feasible region,
$P_{0}$ becomes unstable and the system is uniformly persistent.
Therefore, HIV infection persists. In this case, $P^$* can be
either stable or unstable. We show that $P^$* is stable only for $r$
(the proliferation rate of T cells) small or large and unstable for
some intermediate values of $r.$ In the latter case, numerical
simulations indicate a stable periodic solution exists.

DCDS-B

We discuss the $\omega$-limit set for the Cauchy problem of the porous medium
equation with initial data in some weighted spaces. Exactly, we show that there
exists some relationship between the $\omega$-limit set of the rescaled initial
data and the $\omega$-limit set of the spatially rescaled version of solutions.
We also give some applications of such a relationship.

DCDS-B

In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.

DCDS-B

Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we investigate in a shadow system the effects of migration
and interspecific competition coefficients on the existence of positive solutions. Our study shows that for any given migration, if the interspecific competition coefficient of the invader is
small, then the shadow system has coexistence state; otherwise
we can always find some migration such that it has no coexistence state. Moreover, these
findings can be applied to steady state of a two-species Lotka-Volterra
competition-diffusion model. Particularly, we show that if the interspecific
competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to
coexistence state.

DCDS

In the paper we prove the multiplicity existence of both nonlinear Schrödinger equation and Schrödinger system with slow decaying rate of electric potential at infinity. Namely, for any

, the potentials

have the asymptotic behavior

$\mathit{\boldsymbol{m}},\mathit{\boldsymbol{n > }}{\bf{0}}$ |

$P, Q$ |

$\left\{ \begin{array}{l}P(r) = 1 + \frac{a}{{{r^m}}} + O\left( {\frac{1}{{{r^{m + \theta }}}}} \right),{\rm{ }}\;\;\;\;\theta > 0\\Q(r) = 1 + \frac{b}{{{r^n}}} + O\left( {\frac{1}{{{r^{n + \widetilde \theta }}}}} \right),\;\;\;\;\;\widetilde \theta > 0\end{array} \right.$ |

then Schrödinger equation and Schrödinger system have infinitely many solutions with arbitrarily large energy, which extends the results of [37 ] for single Schrödinger equation and [30 ] for Schrödinger system.

CPAA

We study the dynamics of the one dimensional Swift-Hohenberg
equation defined on a large interval $(-l,l)$ with
Dirichlet-Neumann boundary conditions, where $l>0$ is large and
lies outside of some small neighborhoods of the points $n\pi$ and
$(n+1/2)\pi,n \in N$. The arguments are based on dynamical system
formulation and bifurcation theory. We show that the system with
Dirichlet-Neumann boundary conditions can be reduced to a
two-dimensional center manifold for each bifurcation parameter
$O(l^{-2})$-close to its critical values when $l$ is
sufficiently large. On this invariant manifold, we find families of
steady solutions and heteroclinic connections with each connecting
two different steady solutions. Moreover, by comparing the above
dynamics with that of the Swift-Hohenberg equation defined on $R$
and admitting $2\pi$-spatially periodic solutions in [4],
we find that the dynamics in our case preserves the main features of
the dynamics in the $2\pi$ spatially periodic case.

PROC

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