## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

JIMO

Due to the non-separability of the variance operator, the optimal
investment policy of the multi-period mean-variance model in
Markovian markets doesn't satisfy the time consistency. We propose a
new weak time consistency in stochastic markets and show that the
pre-commitment optimal policy satisfies the weak time consistency at
any intermediate period as long as the investor's wealth is no more
than a specific threshold. When the investor's wealth exceeds the
threshold, the weak time consistency no longer holds. In this case,
by modifying the pre-commitment optimal policy, we derive a wealth
interval, from which we determine a more efficient revised policy.
The terminal wealth obtained under this revised policy can achieve
the same mean as, but not greater variance than those of the
terminal wealth obtained under the pre-commitment optimal policy; a
series of superior investment policies can be obtained depending on
the degree the investor wants the conditional variance to decrease.
It is shown that, in the above revising process, a positive cash
flow can be taken out of the market. Finally, an empirical example
illustrates our theoretical results. Our results generalize existing
conclusions for the multi-period mean-variance model in
deterministic markets.

DCDS

In this
paper we study the mixed initial-boundary value problem for the
equation of time-like extremal surfaces in Minkowski space
$R^{1+(1+n)}$ on the strip $R^{+}\times[0,1]$. Under the
assumptions that the boundary data are small and decaying, we get
the global existence and uniqueness of classical solutions.

DCDS

In this paper we prove the existence of classical solutions to near field reflector problems, both for a point light source and for a parallel light source, with planar receivers. These problems involve Monge-Ampère type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditions by Li, Liu and Nguyen.

DCDS

In this paper we consider the existence and stability of traveling
wave solutions to Cauchy problem of diagonalizable quasilinear
hyperbolic systems. Under the appropriate small oscillation
assumptions on the initial traveling waves, we derive the stability
result of the traveling wave solutions, especially for intermediate traveling waves. As the important examples,
we will apply the results to some systems arising in fluid dynamics
and elementary particle physics.

DCDS-B

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case

$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$ |

where

. There exists a critical mass

found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for

. We obtain global existence of a non-negative entropy weak solution if initial mass is less than

. For

, entropy weak solutions are positive and unique. For

, a finite time blow-up occurs for solutions with initial mass larger than

. For the Cauchy problem with

and initial mass less than

, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or

in

for some subsequence

.

$n≥q 1$ |

$M_c=\frac{2\sqrt{6}π}{3}$ |

$n=1$ |

$M_c$ |

$n≥q 4$ |

$n=1$ |

$M_c$ |

$n=1$ |

$M_c$ |

$ h(·, t_k)\rightharpoonup 0$ |

$L^1(\mathbb{R})$ |

${t_k} \to \infty $ |

KRM

This paper establishes the hyper-contractivity in $L^\infty(\mathbb{R}^d)$
(it's known as ultra-contractivity) for the multi-dimensional
Keller-Segel systems with the diffusion exponent $m>1-2/d$. The
results show that for the supercritical and critical case $1-2/d < m ≤ 2-2/d$, if $||U_0||_{d(2-m)/2} < C_{d,m}$ where $C_{d,m}$ is a
universal constant, then for any $t>0$,
$||u(\cdot,t)||_{L^\infty(\mathbb{R}^d)}$ is bounded and decays as $t$ goes
to infinity. For the subcritical case $m>2-2/d$, the solution
$u(\cdot,t) \in L^\infty(\mathbb{R}^d)$ with any initial data $U_0 \in
L_+^1(\mathbb{R}^d)$ for any positive time.

keywords:
ultra-contractive
,
nonlocal aggregation
,
degenerate diffusion.
,
chemotaxis
,
Hyper-contractive

DCDS-B

This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

DCDS

In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Ampère equation which includes the optimal transportation equation. The estimate extends that in a previous work with Xu-Jia Wang and assumes only that the matrix function in the equation is regular with respect to the gradient variables, that is it satisfies a weak form of the condition introduced previously by Ma,Trudinger and Wang for regularity of optimal transport mappings. We also indicate briefly an application to optimal transportation.

DCDS-B

In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is $C\Delta t$, where $\Delta t$ is the time-step size. Secondly, we prove the convergence rate $C\Delta t^2$ for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate $C\Delta t$.

DCDS

In this paper, we discuss the existence of time quasi-periodic
solutions for the derivative nonlinear Schrödinger equation
$$\label{1.1}\mathbf{i} u_t+u_{xx}+\mathbf{i} f(x,u,\bar{u})u_x+g(x,u,\bar{u})=0
$$
subject to Dirichlet boundary conditions. Using an abstract infinite
dimensional KAM theorem dealing with unbounded perturbation
vector-field and Birkhoff normal form, we will prove that there
exist a Cantorian branch of KAM tori and thus many time
quasi-periodic solutions for the above equation.

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