抛物方程时空动力学

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Preprints:

• (with Jean-Michel Roquejoffre) Large time behaviour in the multidimensional nonlocal Fisher-KPP equation. (pdf)

• (with Weiwei Ding, Zhaoyun Zhang) Bistable traveling fronts in strong shear flows: speed and profile asymptotics. (pdf)

• (with Grégory Faye, Jean-Michel Roquejoffre) Sharp asymptotics for a transport model with a nonlocal condition of the Fisher-KPP type at the boundary. (pdf)

• (with Quentin Griette, François Hamel, Min Zhao) Propagation phenomena in KPP-bistable periodic patchy environments. (pdf) 

• (With Xing Liang, Zhaoyun Zhang) Propagation dynamics for a two-component lattice system with strong competition, preprint.

• Sharp asymptotics for the KPP equation with some front-like initial data. (pdf)

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Highlights:

My work “Sharp asymptotics for the KPP equation with some front-like initial data” develops a unified PDE framework and establishes the first PDE proof of Bramson’s o(1) results in 1983 for two classes of initial tails, with λ* the critical decay rate. To be specific,

    1). Critical exponential decay: xk+1  e−λ*x.

• For k ≥ −3, my analysis, with different PDE method, first recovers the work [Alfaro-Giletti-Xiao], i.e. front locations in O(1) errors and convergence to a bounded family of traveling waves. Beyond that, I further capture the o(1) precision and prove convergence to a single wave, with additional restrictions on the initial tail. 

• For k < −3, my manuscript provides the PDE proof of the o(1) asymptotics, without additional restrictions on the initial tail. The initial data in this regime therefore is called "sufficiently steep". The significance of this part is its role in explaining and comparing the different asymptotic mechanisms.

    2). Subcritical exponential decay: xν e−λx, with 0 < λ < λ* and ν ∈ ℝ.

• My manuscript proves the o(1) results. The previous work [Berestycki-Brunet-Harris-Roberts] for a linear equation with an absorbing boundary provides an important reference. However, to achieve the nonlinear conclusions, additional analysis is mandatory in order for matching lower bounds and profile convergence. In particular, my lower-barrier construction uses an intermediate transformation to address the mismatch between the linear transport scale and the propagation speed. 

N.B. The distinction between O(1) and o(1) is substantive: the latter identifies a limiting profile. The broader importance of o(1) information is illustrated for instance by [Mytnik-Roquejoffre-Ryzhik], where new results on the extremal process of branching Brownian motion is obtained, building on the o(1) information.

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Publications:

• (with Xing Liang, Lei Zhang) The influence of advection on the propagation phenomena of reaction-diffusion equations with KPP-bistable nonlinearity, to appear in Trans. Amer. Math. Soc. (2026). (pdf)

• (with Grégory Faye, Jean-Michel Roquejoffre) Spreading properties in Kermack - McKendrick models with nonlocal spatial interactions – A new look, Comptes Rendus Mathématique, Thematic issue in the memory of Haïm Brezis, 364 (2026), p. 381-429. (pdf)

• (with Samuel Tréton) The influence of an inward-shifting boundary on the heat equation in half-space, SIAM J. Appl. Math., 86 (2026), 1256-1282. (pdf)

• (with François Hamel) KPP transition fronts in a one-dimensional two-patch habitat, Math. Med. Biol., Special issue dedicated to James D. Murray, 42 (2025), 71-97. (pdf)

• (with Christophe Besse, Grégory Faye, Jean-Michel Roquejoffre) The logarithmic Bramson correction for Fisher-KPP equations on the lattice Z, Trans. Amer. Math. Soc. 376 (2023), 8553-8619. (pdf)

• (with François Hamel) Reaction-diffusion fronts in funnel-shaped domains, Adv. Math. 412 (2023), 108807. (pdf)

• (with François Hamel, Frithjof Lutscher) Propagation and blocking in a two-patch reaction-diffusion model, J. Math. Pures Appl. 168 (2022), 213-267. (pdf) 

• (with François Hamel, Frithjof Lutscher) Propagation phenomena in periodic patchy landscapes with interface conditions, J. Dyn. Diff. Equations, In memory of Professor Pavol Brunovský, 36 (2024), 435-486. (pdf) 

• Spreading speeds and pulsating fronts for a field-road model in a spatially periodic habitat, Journal of Differential Equations, 304 (2021), 191-228. (pdf)

• (with Michel Chipot) On some model problem for the propagation of interacting species in a special environment, Discrete Contin. Dyn. Syst. Ser. A,  41 (2021), 3141-3161. (pdf)

• (with Claude-Michel Brauner, Luca Lorenzi) Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface, Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), 581–604. (pdf)

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Seminar and colloquium talks:

2026   PDE Seminar, Shanghai Normal University, May 14.

2026   Nonlinear PDEs and Applications in Mathematical Biology, in honor of Professor Henri Berestycki on the occasion of his 75th birthday, Shanghai Jiao Tong University, April 19-25.

2026   RSME 2026 - Special session: Evolution PDEs in applied sciences, University of Alicante, Alicante, Spain, January 19-23.      

2025   PDE Seminar, Harbin Institute of Technology, Weihai, December 8.

2025   PDE Seminar, Qufu Normal University, November 28.

2025   BIRS Workshop - New rends in Elliptic and Parabolic equations, Banff, Canada, November 16-21.

2025   BIRS Workshop - Emerging Connections between Reaction-Diffusion, Branching Processes, and Biology, Banff, Canada, May 18-23.

2025   ANR Reach Journées 2025, CAMS, EHESS, Paris, April 2-3.

2024   BIRS Workshop - Propagation and stability in evolution equations, UBC Okanagan, Canada, May 26-31.

2024   Orléans Seminar, Institut Denis Poisson, Orléans, April 18.

2024   EDPAN Seminar, LMBP, Université Clermont Auvergne, Clermont-Ferrand, April 4.

2024   LMAH Seminar, Université Le Havre Normandie, Le Havre, March 28.

2024   Journées Jeunes EDPistes en France, Université Paul Sabatier, Toulouse, March 20-22. 

2023   Workshop - Pattern dynamics appearing in reaction-diffusion systems, Meiji University, Tokyo, August 26.

2023   ICIAM, Minisymposium: Pattern dynamics appearing in mathematical biology, Waseda University, Tokyo, August 20-25.

2022   A ReaDiNet seminar day on reaction-diffusion systems in biology, Université de Lorraine, Nancy, November 21.

2022   Workshop on parabolic and kinetic models in population dynamics, Université Paul Sabatier, Toulouse, September 26-30. 

2022   MAC Seminar, Toulouse, May 31.

2021   Young Women in PDEs and Applications, Bonn (Zoom), Germany, September 20-22.

2021   Nonlinear Dynamics Seminar (Fall 2021, MUN), Memorial University of Newfoundland (Zoom), Canada, September 19.

2020   Nonlinear Dynamics Seminar (Fall 2020, MUN), Memorial University of Newfoundland (Zoom), Canada, November 27.

2020   Nonlinear Dynamics Seminar (winter 2020, MUN), Memorial University of Newfoundland (Zoom), Canada, April 7.&14.

2019   International Conference on Elliptic and Parabolic Problems, Gaeta, Italy, May 20-24.