Special Session 31: Regularity of partial differential equations

Dynamical behaviors of a classical competition system with advection

Yanling Li Shaanxi Normal University Peoples Rep of China Co-Author(s):    Yanling Li

Abstract: This paper deals with a two-species competition model in a homogeneous advective environment, where two species are subjected to a net loss of individuals at the downstream end. Under the assumption that the advection and diffusion rates of two species are proportional, we give a basic classification on the global dynamics by employing the theory of monotone dynamical system. It turns out that bistability does not happen, but coexistence and competitive exclusion may occur. Furthermore, we present a complete classification on the global dynamics in terms of the growth rates of two species. However, once the above assumption does not hold, bistability may occur. In detail, there exists a tradeoff between growth rates of two species such that competition outcomes can shift between three possible scenarios, including competitive exclusion, bistability and coexistence. These results show that growth competence is important to determine dynamical behaviors.

Interior pointwise regularity for elliptic and parabolic equations in divergence form and applications to nodal sets

Yuanyuan Lian Department of Mathematical Analysis, University of Granada Spain Co-Author(s):

Abstract: In this talk, we obtain the interior pointwise $C^{k,\alpha}$ ($k\geq 0$, $ 0 <\alpha< 1 $) regularity for weak solutions of elliptic and parabolic equations in divergence form. The compactness method and perturbation technique are employed. The pointwise regularity is proved in a very simple way and the results are optimal. In addition, these pointwise regularity can be used to characterize the structure of the nodal sets of solutions.

The dynamical behavior and coexistence of a predator-prey model in the chemostat

Jianhua Wu Shaanxi Normal University Peoples Rep of China Co-Author(s):

Abstract: This paper deals with a diffusive predator-prey system in the chemostat which describes the growth of planktonic rotifers, feeding on unicellular green algae. The dynamical behavior and coexistence of this system is characterized by using the diffusion rate.

Log BMO matrix weights and quasilinear elliptic equations with Orlicz growth

Rui Yang Central South University Peoples Rep of China Co-Author(s):    Sun-Sig Byun

Abstract: We study a very general quasilinear elliptic equation with the nonlinearity with Orlicz growth subject to a degenerate or singular matrix-valued weight on a bounded nonsmooth domain. The nonlinearity satisfies a nonstandard growth condition related to the associated Young function, and the logarithm of the matrix-valued weight in BMO is constrained by a smallness parameter which has a close relationship with the Young function. We establish a global Calder\`{o}n-Zygmund estimate for the weak solution of such a degenerate or singular problem in the setting of a weighted Orlicz space under a minimal geometric assumption that the boundary of the domain is sufficiently flat in the Reifenberg sense.

Boundary regularity for elliptic equations

Kai Zhang University of Granada Spain Co-Author(s):    Yongpan Huang, Guanghao Hong, Dongsheng Li, Yuanyuan Lian, Duan Wu

Abstract: In this talk, we introduce a series of boundary pointwise regularity for elliptic equations, including boundary Hölder regularity, boundary Lipschitz regularity, boundary differentiability, boundary $C^{k,\alpha}$ regularity for any $k\geq 1, 0 < \alpha < 1$. This talk is a combination of our several work in recent years.

Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

Zhengce Zhang Xi`an Jiaotong University Peoples Rep of China Co-Author(s):    Caihong Chang, Bei Hu

Abstract: In this talk, we consider two properties of positive weak solutions of quasilinear elliptic equations, $-\Delta_{m}u=u^q|\nabla u|^p\ \mathrm{in}\ \mathbb{R}^N$, with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the $m$-Laplacian operator. The technique of Bernstein gradient estimates is ultilized to study the case $p