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Mathematical Analysis and Modelling Research Group

About Us

This group develops theoretical techniques through analytical as well as topological and algebraic tools, and apply the ideas and results obtained in the theoretical investigation to problems arising from a wide range of areas, including various important models in differential geometry and natural sciences (biology, chemistry, ecology, physics and rural sciences). The concrete problems we currently work on include the Chern-Simons model in condensed matter physics, the logistic and related models in population biology. Important problems in these applied areas serve as a key source of the topics for investigation by the group.

Research Topics

  • Nonlinear partial differential equations
  • Reaction-diffusion equations
  • Mathematical biology
  • Nonlinear functional analysis
  • Complex analysis
  • CR-Geometry
  • Complex analytic geometry
  • Pseudoholomorphic curves
  • Ecological statistics
  • Statistical modelling

Staff Researchers

Postgraduate Students

  • Kumbu Dorji (PhD)
  • Masoud Ganjiarjenaki (PhD)
  • Benniao Li (PhD)
  • Reginald Page (PhD)
  • Ben Mclaughlin (PhD)

External Collaborators

  • Dr Florica Cirstea (Univ. of Sydney)
  • Prof Daomin Cao (Chinese Academy of Sci., China)
  • Prof Norman Dancer (Univ. of Sydney)
  • Prof Vladimir Ezhov (Univ. of South Australia)
  • Dr Timur Gureyev (CSIRO - Manufacturing and Infrastructure Technology, Melbourne)
  • Prof Zongming Guo (Henan Normal Univ., China)
  • Prof Sze-Bi Hsu (National Tsinghua Univ., Taiwan)
  • Prof Kang-Tae Kim (POSTECH, Rep. of Korea)
  • Prof Martin Kolar (Masaryk Univ., Czech Rep.)
  • Prof Xing Liang (Chinese Univ of Sci and Tech, China)
  • Prof Changshou Lin (National Taiwan Univ.)
  • Prof Zhigui Lin (Yangzhou Univ., China)
  • Prof. Bendong Lou (Tongji University, China)
  • Dr Gabriel Paternain (Cambridge Univ., UK)
  • Dr Rui Peng (Memorial Univ of Newfoundland, Canada)
  • Prof Juncheng Wei (Chinese Univ. of Hong Kong)

Research Projects and Grants Since 2004

  1. Maolin Zhou, Nonlinear free boundary problems: propagation and regularity, ARC DECRA award, 2017-2019, $330,324.
  2. S. Yan, E.N. Dancer, Y. Du and C.S. Lin, Nonlinear partial differential equations: bubbles, layers and stability, ARC discovery grant, 2017-2019, $345,000.
  3. Y. Du, Propagation described by partial differential equations with free boundary, ARC discovery grant, 2015-2018, $415,000.
  4. S. Yan, E.N. Dancer and Y. Du, Singularity, degeneracy and related problems in nonlinear partial differential equations, ARC discovery grant, 2013-2015, $300,000.
  5. G. Schmalz, Y. Nikolayevsky, G. Cairns, V. Ejov and D. Alekseevsky, Symmetries in real and complex geometry, ARC discovery grant, 2013-2015, $258,000.
  6. Y. Du, Propagation and free boundary problems in nonlinear partial differential equations, ARC discovery grant, 2012-2014, $255,000.
  7. Y. Du, E.N. Dancer and S. Yan, Transitions and Singular Behaviour in Nonlinear Partial Differential Equations, ARC discovery grant, 2010-2012, $306,000.
  8. Y. Du, Mathematical modelling of the spreading of invasive species, UNE PVCR grant, 2010, $30,000.
  9. Y. Du, S, Yan, G. Schmalz and A. Harris, Singularities in nonlinear and complex analysis, UNE IRL grant, 2009, $13,000.
  10. Y. Du, E.N. Dancer and S. Yan, Sharp Transitions in Partial Differential Equations and Related Problems, ARC discovery grant, 2007-2009, $240,000.
  11. Y. Du, A. Harris, G. Schmalz and S. Yan, Partial differential equations and complex geometry, UNE grant, 2007-2008, $11,000
  12. Y. Du and E.N. Dancer, Free Boundary Problems in Partial Differential Equations and Related Topics, ARC discovery grant, 2003-2006, $158,000.
  13. V. Ejov, G. Schmalz and A Spiro, Normal forms and Chern-Moser Connection in the Study of Cauchy-Riemann Manifolds, ARC discovery grant, 2003-2006, $165,000.
  14. Y. Du, Sharp transitions in partial differential equations and related problems, UNE RPG, 2006, $14,500.
  15. Y. Du, Singular perturbation and transition layers in heterogeneous environment, Australian Academy of Science and JSPS (Japan), 2005, $7,600.
  16. Y. Du, Partial differential equations with unconventional boundary conditions and applications, UNE RPG, 2005, $13,000.
  17. Y. Du, Partial differential equations with unconventional boundary conditions and applications, UNE RPG, 2004, $10,000.

Publications since 2007

Books

  1. Y. Du, Hitoshi Ishii and Wei-Yueh Lin (editors), Recent Progress in Nonlinear Partial Differential Equations and Viscosity Solutions, World Scientific Publishing, 372 pages. ISBN 981-283-473-7, 2009.
  2. Y. Du (author), Order structure and topological methods in nonlinear partial differential equations. Vol. 1, Maximum principles and applications, Series in Partial Differential Equations and Applications, 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, x+190 pages. ISBN 981-256-624-4, 2006.

Papers

  1. Deng, Y.; Lin, C.-S.; Yan, S.: On the prescribed scalar curvature problem in R^N, local uniqueness and periodicity, preprint, 2014.
  2. Lin, Chang-Shou; Yan, Shusen: On condensate of solutions for the Chern-Simons-Higgs equation, preprint, 2014.
  3. Lin, Chang-Shou; Yan, Shusen: On the Chern-Simons-Higgs equation: Part II, local uniqueness and exact number of solutions, preprint, 2014.
  4. 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.
  5. Joo, Jae-Cheon; Kim, Kang-Tae; Schmalz, Gerd: A generalization of Forelli's theorem. Math. Ann. 355 (2013), no. 3, 1171-1176.
  6. Lin, Chang-Shou; Yan, Shusen: Bubbling solutions for the SU(3) Chern-Simons model on a torus. Comm. Pure Appl. Math. 66 (2013), no. 7, 991-1027.
  7. Lin, Chang-Shou; Yan, Shusen: Existence of bubbling solutions for Chern-Simons model on a torus. Arch. Ration. Mech. Anal. 207 (2013), no. 2, 353-392.
  8. Yan, Shusen; Yang, Jianfu Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents. Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 587–610.
  9. Cao, Daomin; Peng, Shuangjie; Yan, Shusen: On the Webster scalar curvature problem on the CR sphere with a cylindrical-type symmetry. J. Geom. Anal. 23 (2013), no. 4, 1674-1702.
  10. Wei, Juncheng; Yan, Shusen: Infinitely many nonradial solutions for the Henon equation with critical growth. Rev. Mat. Iberoam. 29 (2013), no. 3, 997-1020.
  11. 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.
  12. E.N. Dancer, Y. Du and M. Efendiev, Quasilinear elliptic equations on half- and quarter-spaces, Adv. Nonlinear Studies (special issue dedicated to Klaus Schmitt), 13 (2013), 115-136.
  13. Y. Du and Z.M. Guo, Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Diff. Eqns., 18(2013), 737-768.
  14. Y. Du and R. Peng, Sharp spatiotemporal patterns in the diffusive time-periodic logistic equation, J. Diff. Eqns., 254 (2013), 3794-3816.
  15. A. Radu, T. Radu, C. McGraw, P. Dillingham, S. Anastasova-Ivanova, D. Diamond (2013). Ion selective electrodes in environmental analysis. Journal of the Serbian Chemical Society 78:1729-1761.
  16. J.E. Moore, K.A. Curtis, R.L. Lewison, P.W. Dillingham, J.M. Cope, S.V. Fordham, S.S. Heppell, S.A. Pardo, C.A. Simpfendorfer, G.N. Tuck, S. Zhou (2013). Evaluating sustainability of fisheries bycatch mortality for marine megafauna: a review of conservation reference points for data-limited populations. Environmental Conservation 40:329-344.
  17. P.W. Dillingham, T. Radu, D. Diamond, A. Radu, C.M. McGraw (2012). Bayesian Methods for Ion Selective Electrodes. Electroanalysis 24:316-324
  18. K. Hageman, K. Lavin, S. Marx, P. Dillingham, B. Kamber (2012). Using trace elements in particulate matter to identify the sources of semi-volatile organic contaminants in air at an alpine site. Environmental Science & Technology 46:268-276.
  19. P.W. Dillingham, G.P. Elliott, K.J. Walker, D. Fletcher (2012). Adjusting age at first breeding of albatrosses and petrels for emigration and study duration. Journal of Ornithology 153:205-217.
  20. G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media (special issue dedicated to H. Matano), 7 (2012), 583-603.
  21. Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies, Trans. Amer. Math. Soc., 364(2012), 6039-6070. (A*)
  22. Y. Du and Z.M. Guo, The Stefan problem for the Fisher-KPP equation, J. Diff. Eqns., 253(2012), 996-1035. (A*)
  23. Y. Du and Li Ma, A Liouville theorem for conformal Gaussian curvature type equations in $\R^2$, Calculus of Variations and PDEs, 43(2012), 485-505. (A*)
  24. Cao, Daomin; Peng, Shuangjie; Yan, Shusen Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth. J. Funct. Anal. 262 (2012), no. 6, 2861-2902. (A*)
  25. Chen, Wenyi; Wei, Juncheng; Yan, Shusen Infinitely many solutions for the Schrodinger equations in RN with critical growth. J. Differential Equations 252 (2012), no. 3, 2425-2447. (A*)
  26. Schmalz, Gerd; Slovak, Jan: Free CR distributions. Cent. Eur. J. Math. 10 (2012), no. 5, 1896-1913.
  27. Y. Du and Linfeng Mei, On a nonlocal reaction-diffusion-advection equation modeling phytoplankton, Nonlinearity, 24(2011), 319-349. (A*)
  28. Y. Du and Zongming Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary II, J. Diff. Eqns., 250(2011), 4336-4366. (A*)
  29. Wei, Juncheng; Yan, Shusen Infinitely many positive solutions for an elliptic problem with critical or supercritical growth. J. Math. Pures Appl. (9) 96 (2011), no. 4, 307-333. (A*)
  30. Cao, Daomin; Yan, Shusen Infinitely many solutions for an elliptic Neumann problem involving critical Sobolev growth. J. Differential Equations 251 (2011), no. 6, 1389-1414.(A*)
  31. Wang, Liping; Wei, Juncheng; Yan, Shusen On Lin-Ni's conjecture in convex domains. Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1099-1126.(A*)
  32. Ezhov, Vladimir; McLaughlin, Ben; Schmalz, Gerd From Cartan to Tanaka: getting real in the complex world. Notices Amer. Math. Soc. 58 (2011), no. 1, 20-27.
  33. Baues, Hans-Joachim; Bleile, Beatrice Self-maps of the product of two spheres fixing the diagonal. Topology Appl. 158 (2011), no. 16, 2198-2204.
  34. Harris, Adam; Kolar, Martin On infinitesimal deformations of the regular part of a complex cone singularity. Kyushu J. Math. 65 (2011), no. 1, 25-38.
  35. Cao, Daomin; Peng, Shuangjie; Yan, Shusen Multiplicity of solutions for the plasma problem in two dimensions. Adv. Math. 225 (2010), no. 5, 2741-2785.(A*)
  36. F. Cirstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations, J. Functional Anal., 259, pp 174-202, 2010. (A*)
  37. Y. Du and Zhigui Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42(2010), 377-405. (A)
  38. Y. Du and Hiroshi Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. European Math. Soc., 12(2010), 279-312. (A*)
  39. Wei, Juncheng; Yan, Shusen On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 2, 423-457. (A*)
  40. Lin, Chang-Shou; Yan, Shusen Bubbling solutions for relativistic abelian Chern-Simons model on a torus. Comm. Math. Phys. 297 (2010), no. 3, 733-758.(A*)
  41. Cao, Daomin; Yan, Shusen Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential. Calc. Var. Partial Differential Equations 38 (2010), no. 3-4, 471-501.(A*)
  42. Wang, Liping; Wei, Juncheng; Yan, Shusen A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture. Trans. Amer. Math. Soc. 362 (2010), no. 9, 4581-4615.(A*)
  43. Wei, Juncheng; Yan, Shusen Infinitely many solutions for the prescribed scalar curvature problem on S^N. J. Funct. Anal. 258 (2010), no. 9, 3048-3081.(A*)
  44. Wei, Juncheng; Yan, Shusen Infinitely many positive solutions for the nonlinear Schröger equations in RN. Calc. Var. Partial Differential Equations 37 (2010), no. 3-4, 423-439.(A*)
  45. J. Schmalz, G. Schmalz, T. Gureyev, and K. Pavlov, On the derivation of the Greens function for the Helmholtz equation using generalized functions, Am. J. Phys., (2010), Vol. 78, No. 2, pp. 181-186. (B)
  46. Bleile, Beatrice Poincare duality pairs of dimension three. Forum Math. 22 (2010), no. 2, 277-301.
  47. Albert, C.; Bleile, B.; Frohlich, J. Batalin-Vilkovisky integrals in finite dimensions. J. Math. Phys. 51 (2010), no. 1, 015213, 31 pp.
  48. Y. Du and Yoshio Yamada, On the long-time limit of positive solutions to the degenerate logistic equation, Discrete and Continuous Dynamical Systems A, 25(2009), 123-132. (A)
  49. Y. Du, Rui Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Eqns., 246, pp. 3932-3956, 2009. (A*)
  50. Y. Du and Z.M. Guo, Positive solutions of an elliptic equation with negative exponent: stability and critical power, J. Diff. Eqns., 246, pp. 2387-2414, 2009. (A*)
  51. Y. Du, Change of environment in model ecosystems: effect of a protection zone in diffusive population models, Recent Progress in Nonlinear Partial Differential Equations and Viscosity Solutions, Y. Du, H. Ishii and W.Y. Lin (eds.), World Scientific Publishing, ISBN 978-981-283-473-7, 2009.
  52. Cao Daomin, Peng Shuangjie and Yan Shusen, Asymptotic behaviour of ground state solutions for the Henon equation, IMA J. Appl. Math., 74, no. 3, pp. 468-480, 2009. (A)
  53. Dancer Edward N., Hilhorst Danielle and Yan Shusen, Peak solutions for the Dirichlet problem of an elliptic system, Discrete Contin. Dyn. Syst. no. 3, pp. 731-761, 2009. (A)
  54. K.T.Kim, E.Poletsky and G.Schmalz, Functions holomorphic along holomorphic vector fields, Journal of Geometric Analysis, 19 (2009), no. 3, 655-666. (A)
  55. V. Ejov, M. Kolar and G. Schmalz, Degenerate hypersurfaces with a two-parametric family of automorphisms, Complex Variables and Elliptic Equations, 54 (2009), no. 3-4, 283-291. (B)
  56. Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasiliear problem modelling phytoplankton I: Existence, SIAM J. Math. Anal., 40, pp. 1419-1440, 2008. (A)
  57. Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasiliear problem modelling phytoplankton II: Limiting profile, SIAM J. Math. Anal., 40, pp. 1441-1470, 2008. (A)
  58. Y. Du, P.Y.H. Pang and M. Wang, Qualitative analysis of a predator-prey model with stage structure for the predator, SIAM J. Appl. Math., 69, pp. 596-620, 2008. (A*)
  59. Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Eqns., 244, pp. 61-86, 2008. (A*)
  60. Y. Du, The heterogeneous Allen-Cahn equation in a ball: solutions with layers and spikes, J. Diff. Eqns., 244, pp. 117-169, 2008. (A*)
  61. Y. Du, Z. Liu, A. Pistoia and Shusen Yan, Sign changing solutions with clustered layers near the origin for singularly perturbed semilinear elliptic problems on a ball, Methods and Applications of Analysis (special issue dedicated to N.S. Trudinger), 15, pp. 137-148, 2008.
  62. A. Harris and M. Kolar, A Remark on cohomology with supports in the complement of a cone singularity, RIMS Kokyuroku, 1610, pp. 32-37, 2008.
  63. A. Harris and G. Paternain, Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders, Ann. Global Anal and Geom., 34, pp. 115-134, 2008. (A)
  64. A. Harris and K. Wysocki, Branch-structure of J-holomorphic curves near periodic orbits of a contact manifold, Trans. Amer. Math. Soc., 360, pp. 2131-2152, 2008. (A*)
  65. V. Beloshapka, V. Ejov and G. Schmalz, Holomorphic classification of 4-dimensional surfaces in C3, Izv. Ross. Akad. Nauk Ser. Mat., 72:3, pp. 3-18, 2008. (B)
  66. V. Ejov, G. Schmalz and A. Spiro, CR-manifolds of codimension two of parabolic type, Indiana University Mathematics Journal, 57:1, pp. 309-342, 2008. (A)
  67. C.-K. Han, J.-W. Oh and G. Schmalz, Symmetry algebra for multi-contact structures given by 2n vector fields on ?, Mathematische Annalen, 341:3, pp. 529-542, 2008. (A*)
  68. Cao Daomin, Noussair Ezzat S. and Yan Shusen, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Trans. Amer. Math. Soc., 360, no. 7, pp. 3813-3837, 2008. (A*)
  69. Dancer EN and Yan S, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, Journal of the London Mathematical Society, 78, pp. 639-662, 2008. (A)
  70. Dancer EN and Yan S, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Methods and Applications of Analysis, 15, no. 1, pp. 97-120, 2008. (B)
  71. Baues, Hans Joachim; Bleile, Beatrice Poincare duality complexes in dimension four. Algebr. Geom. Topol. 8 (2008), no. 4, 2355-2389.
  72. F. Cirstea and Y. Du, Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity, J. Functional Anal., 250, pp. 317-346, 2007. (A*)
  73. Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation, J. Diff. Eqns., 238, pp. 87-117, 2007. (A*)
  74. F. Cirstea and Y. Du, Large solutions of elliptic equations with a weakly superlinear nonlinearity, J. d'Analyse Math., 103, pp. 261-277, 2007. (A)
  75. Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359, pp. 4557-4593, 2007. (A*)
  76. Y. Du, Z.M. Guo and F. Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem, Discrete and Continuous Dynamical Systems, 19, pp. 271-298, 2007. (A)
  77. V. Ezhov and G. Schmalz, Elliptic CR-manifolds and shear invariant ODE with additional symmetries, Arkiv Matematik, 45:2, pp. 253-268, 2007. (B)
  78. V. Beloshapka, V. Ejov and G. Schmalz, Canonical Cartan connection and holomorphic invariants on Engel CR manifolds, Russ. J. Math. Phys., 14:2, pp. 121-133, 2007.
  79. Dancer E.N. and Yan S., Remarks on the existence of many solutions of certain nonlinear elliptic equations, Bolletino U.M.I., 8, no. 10 (B), pp. 1013-1023, 2007. (B)
  80. Li Gongbao, Peng Shuangjie and Yan Shusen, A new type of solutions for a singularly perturbed elliptic Neumann problem, Rev. Mat. Iberoam, 23, no. 3, pp. 1039-1066, 2007.
  81. Wei Juncheng and Yan Shusen, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth, J. Math. Pures Appl., (9) 88, no. 4, pp. 350-378, 2007. (A*)
  82. Wei Juncheng and Yan Shusen, New solutions for nonlinear Schrodinger equations with critical nonlinearity, J. Differential Equations, 237, no. 2, pp. 446-472, 2007. (A*)
  83. Dancer Edward Norman, Ren Xiaofeng and Yan Shusen, On multiple radial solutions of a singularly perturbed nonlinear elliptic system, SIAM J. Math. Anal., 38, no. 6, pp. 2005-2041, 2007. (A)
  84. Wei Juncheng and Yan Shusen, Lazer-McKenna conjecture: the critical case, J. Funct. Anal., 244, no. 2, pp. 639-667, 2007. (A*)
  85. Li Gongbao, Yan Shusen and Yang Jianfu, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations, 28, no. 4, pp. 471-508, 2007. (A*)
  86. Dancer E. N. and Yan Shusen, A new type of concentration solutions for a singularly perturbed elliptic problem, Trans. Amer. Math. Soc., 359, no. 4, pp. 1765-1790, 2007. (A*)

Events

Contact us

    Professor Yihong Du
    School of Science and Technology
    University of New England, Armidale, NSW 2351
    Email: ydu@turing.une.edu.au
    Phone: 02 6773 3066 (or +61 2 6773 3066 overseas)
    Homepage: http://turing.une.edu.au/staff/~ydu