GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.

ATTRACTORS FOR THE GINZBURG-LANDAU-BBM EQUATIONS IN AN UNBOUNDED DOMAIN

By the interpolating inequality and a priori estimates in the weighted space,the existence of the global solutions for the Ginzburg-Landau equation coupled with the BBM equation in an unbounded domain is considered, and the existence of the maximal attractor is obtained.

Further study of a weighted elliptic equation

A Liouville type result is established for non-negative entire solutions of a weighted elliptic equation.This provides a positive answer to a problem left open by Du and Guo(2015) and Phan and Souplet(2012)(see(CJ) by Du and Guo(2015) and Conjecture B by Phan and Souplet(2012)). Meanwhile, some regularity results are also obtained. The main results in this paper imply that the number ps is the critical value of the Dirichlet problems of the related equation, even though there are still some open problems left. Our results also apply for the equation with a Hardy potential.

A Visual Attention Model for Robot Object Tracking

Inspired by human behaviors, a robot object tracking model is proposed on the basis of visual attention mechanism, which is fit for the theory of topological perception. The model integrates the image-driven, bottom-up attention and the object-driven, top-down attention, whereas the previous attention model has mostly focused on either the bottom-up or top-down attention. By the bottom-up component, the whole scene is segmented into the ground region and the salient regions. Guided by top-down strategy which is achieved by a topological graph, the object regions are separated from the salient regions. The salient regions except the object regions are the barrier regions. In order to estimate the model, a mobile robot platform is developed, on which some experiments are implemented. The experimental results indicate that processing an image with a resolution of 752 × 480 pixels takes less than 200 ms and the object regions are unabridged. The analysis obtained by comparing the proposed model with the existing model demonstrates that the proposed model has some advantages in robot object tracking in terms of speed and efficiency.

Weighted Pseudo Almost Periodic Solutions of N-th Order Neutral Differential Equations with Piecewise Constant Arguments

In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations.

A class of weighted estimating equations for additive hazards models with covariates missing at random

Missing covariate data arise frequently in biomedical studies.In this article,we propose a class of weighted estimating equations for the additive hazards regression model when some of the covariates are missing at random.Time-specific and subject-specific weights are incorporated into the formulation of weighted estimating equations.Unified results are established for estimating selection probabilities that cover both parametric and non-parametric modelling schemes.The resulting estimators have closed forms and are shown to be consistent and asymptotically normal.Simulation studies indicate that the proposed estimators perform well for practical settings.An application to a mouse leukemia study is illustrated.

Weighted Pseudo Asymptotically Periodic Mild Solutions of Evolution Equations

In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.

Qualitative properties and classification of solutions to elliptic equations with Stein-Weiss type convolution part

In this paper,we study the qualitative properties and classification of the solutions to the elliptic equations with Stein-Weiss type convolution part.Firstly,we study the qualitative properties,such as the symmetry,regularity and asymptotic behavior of the positive solutions.Secondly,we classify the non-positive solutions by proving some Liouville type theorems for the finite Morse index solutions and stable solutions to the nonlocal elliptic equations with double weights.

Additive Hazards Regression under Generalized Case-Cohort Sampling

Case-cohort design usually requires the disease rate to be low in large cohort study,although it has been extensively used in practice.However,the disease with high rate is frequently observed in many clinical studies.Under such circumstances,it is desirable to consider a generalized case-cohort design,where only a fraction of cases are sampled.In this article,we propose the inference procedure for the additive hazards regression under the generalized case-cohort sampling.Asymptotic properties of the proposed estimators for the regression coefcients are established.To demonstrate the efectiveness of the generalized case-cohort sampling,we compare it with simple random sampling in terms of asymptotic relative efciency.Furthermore,we derive the optimal allocation of the subsamples for the proposed design.The fnite sample performance of the proposed method is evaluated through simulation studies.