Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations

We prove a local existence and uniqueness theorem for abstract parabolic problems of the type when the nonlinearity satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.

     

Totally umbilic null hypersurfaces in generalized Robertson–Walker spaces

We show that there is a correspondence between totally umbilic null hypersurfaces in generalized Robertson–Walker spaces and twisted decompositions of the fibre. This allows us to prove that nullcones are the unique totally umbilic null hypersurfaces in the closed Friedmann Cosmological model. We also apply this kind of ideas to static spaces, in particular to Reissner–Nordström and Schwarzschild exterior spacetimes.     

Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.).     

Topic tracking with Bayesian belief network

The task of topic tracking is to monitor a stream of stories and find all subsequent stories that discuss the same topic. Using Bayesian belief network we give three topic tracking models: a static topic model BSTM and two dynamic topic models BDTM-I, BDTM-II. BDTM-II merges the advantages of BSTM and BDTM-I, has better tracking performance than the former two, and effectively alleviates topic drift phenomenon. Applying unrelated coming stories to update BDTM-I and BDTM-II can filter noises existed in topics. Experiments on TDT corpora show that BSTM decreases (C_det)_norm by 5.5% comparing to VSM, BDTM-II decreases (C_det)_norm by 6.3% and 6.0% comparing to BSTM and BDTM-I respectively, using unrelated stories can improve the tracking performance.     

Optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities

In this paper, optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities are investigated via Darboux transformation. We derive family of structures of rogue wave, including rogue waves with one peak and two valleys, bright rogue waves without valleys while with one peak or two peaks.     

Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions

In this paper, we consider the following quasilinear elliptic exterior problem

     

含有陡峭势阱和凹凸非线性项的Kirchhoff型问题的多重正解*

在这篇文章中, 作者研究涉及凹凸非线性项的Kirchhoff型问题-(a + b ∫R3|▽u|²dx) Δu + λV (x)u = μf(x)|u|^q?2u + |u|^p?2u, x ∈ R³,u ∈ H¹(R³),其中a,b > 0 是常数, λ, μ > 0 是参数, 1 < q < 2, 4 < p < 6 且 V 是一个非负连续位势. 在f(x) 和 V 的合适条件下,此问题正解的存在性和集中性能够通过Nehari 流形和Ekeland 变分原理得到.