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2021, Vol. 41 No. 3   Published date: 30 September 2021
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• Extension of Quasi-plurisubharmonic Functions

  2021, 41(3): 1-12. doi:

  Abstract ( 1432 )   PDF (8763KB) ( 695 )   　　

  In this paper, we give a survey on the extension of (quasi-)plurisubharmonic functions from complex submanifolds.

  We firstly review the extension of plurisubharmonic functions on Stein manifolds, and then review the extension of quasi-plurisubharmonic functions on compact complex manifolds, including some unpublished new results of the authors Wang and Zhou.

• Mean Field Equation on Spheres

  2021, 41(3): 13-37. doi:

  Abstract ( 1826 )   PDF (16655KB) ( 1198 )   　　

  In this expository note, we will introduce the recent progress and open problems concerning mean field type equations on spheres. In particular, some new inequalities of Aubin-Onofri type as well as their close connection to mean field type equations are presented. 

• Unique Continuation from a Vertex Corner for Maxwell's System

  2021, 41(3): 38-58. doi:

  Abstract ( 1395 )   PDF (10779KB) ( 448 )   　　

  In this paper, we establish a novel unique continuation property (UCP) for the Maxwell system locally around a vertex corner. It is shown that if certain homogeneous conditions are imposed on the planes forming the vertex corner, the electric/magnetic field must vanish to a certain order. We derive the vanishing order and relate it to the polyhedral angles of the vertex corner. This extends the recent results in [21] where the UCP was considered associated with an edge corner. This type of UCP study originated from the study of a longstanding problem in inverse scattering theory. 

• Some Research on Limit Cycles of Li\'enard System

  2021, 41(3): 59-95. doi:

  Abstract ( 1840 )   PDF (24586KB) ( 710 )   　　

  The aim of this paper is to introduce the progress on the research for limit cycles of Li\'enard systems and present some new results. The results focus on four problems: the existence of limit cycles, the uniqueness of limit cycles, the exact number of limit cycles and the upper bound of limit cycles. Finally, we summarize some methods for studying limit cycles of Li\'enard systems.

• Explicit High-order Maximum Principle Preserving Schemes for the Conservative Allen--Cahn Equation

  2021, 41(3): 96-110. doi:

  Abstract ( 1856 )   PDF (11270KB) ( 353 )   　　

  Compared with the well-known classical Allen--Cahn equation, the modified Allen--Cahn equation, equipped with a nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this paper, a class of up to eighth-order maximum principle preserving schemes are proposed for solving the modified conservative Allen--Cahn equation. Based on the second-order finite-difference space discretization, we investigate the high-order integrating factor two-step Runge--Kutta maximum principle preserving schemes. We prove that the schemes can preserve the maximum principle and mass of the conservative Allen--Cahn equation and give the convergence analysis of proposed schemes. Finally, two- and three-dimensional numerical tests are carried out to verify the theoretical results and demonstrate the performance of proposed schemes.

• Analysis of Dynamics in Toxin-phytoplankton-zooplankton Reaction-diffusion Model with Prey-taxis

  2021, 41(3): 111-129. doi:

  Abstract ( 1393 )   PDF (8080KB) ( 526 )   　　

  This paper investigates a time-delayed reaction-diffusion model with chemotaxis for phytoplankton. The stability of the positive steady states is obtained by analyzing the related characteristic equations. With the help of Crandall- Rabinowitz's local partial bifurcation theory, the chemotaxis sensitivity coefficient and delay are taken as bifurcation parameters respectively to investigate the existence of Turing bifurcation and Hopf bifurcation. Then, the direction and stability of Hopf bifurcation are studied by using the central manifold theorem and the normal form method. Finally, numerical simulations are presented to show the influence of chemotaxis and time delay on the bifurcation and pattern formation of the system. Our results show that: in the system without time delay, when the chemotaxis sensitivity coefficient exceeds a critical value, the positive steady-state solution of the system will change from stable to unstable (Turing instability); in the system with delay, when the chemotaxis sensitivity coefficient is less than the critical value, if the delay is below a certain value, the positive steady-state solution is locally asymptotically stable; if the time delay exceeds a certain value, then the system will undergo Hopf bifurcation at the positive steady-state solution, and a stable spatially homogeneous periodic solution will be bifurcated from the positive steady-state solution.