Propagation Described by Partial Differential Equations with Free Boundary

Project title: Propagation Described by Partial Differential Equations with Free Boundary

Principal investigator: Prof. Yihong Du (UNE)

Other participants:
Dr Maolin Zhou (UNE), Dr Weiwei Ding (UNE), Prof. Bendong Lou (Shanghai Normal Univ), Prof Xing Liang (Univ of Sci and Tech of China)

Funding body:   Australian Research Council (2015-2018)

Project description:
Front propagation appears in many branches of sciences; examples include the propagation of nerve impulses, the propagation of premixed flames, the spreading of advantageous genes, and the spreading of an invasive species. Although the sources of propagation are diverse, the basic phenomena have been observed to follow certain rules that can be captured by mathematical models of nonlinear partial differential equations (PDEs) or systems of such equations. Thus it is possible for a mathematical theory based on a set of such equations to provide vital insights and useful predictions for the spreading of a variety of subjects arising in different areas of sciences. This topic has been investigated by many leading mathematicians in the past several decades, and extensive research in this area is conducted by several first rate research groups around the world. With the advances of the relevant sciences, more and more demanding questions arise in this area, and many important and basic questions still remain open.  The aim of this project is to develop new mathematics for better applications and deeper insights into the propagation phenomena. We modify existing models by introducing a free boundary to represent the spreading front, and so the partial differential equations are satisfied over a varying spatial domain whose boundary evolves with time, which need to be solved together with the partial differential equation. This makes the model more realistic but at the same time makes the mathematical treatment of the model much more difficult to handle. This research is mostly theoretical though numerical simulations are also needed from time to time.

Related publications:

  1. Y. Du, Mingxin Wang and Maolin Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pure Appl., 107(2017), 253-287.
  2. Weiwei Ding, Y. Du and Xing Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions, J. Diff. Eqns., 262(2017), 4988-5021.
  3. Y. Du, Hiroshi Matsuzawa and Maolin Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103(2015), 741-787.
  4. Y. Du and Xing Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32(2015), 279-305.
  5. Y. Du and Bendong Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17(2015), 2673-2724.
  6. Y. Du, Hiroshi Matsuzawa and Maolin Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46(2014), 375-396.
  7. Y. Du, Hiroshi Matano and Kelei Wang, Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Rational Mech. Anal., 212(2014), 957-1010.
  8. Y. Du and Zhigui Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Cont. Dyn. Syst.-B, 19(2014), 3105-3132.
  9. Y. Du, Zongming Guo and Rui Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
  10. Y. Du, Establishment or vanishing: fate of an invasive species based on mathematical models, in "The Balance of Nature and Human Impact", edited by Klaus Rohde, Cambradge Univ Press, 2013, pp231-238.