Juncheng Wei


Research Classification

Research Interests

applied and geometric analysis
Mathematical biology
Nonlinear partial differential equations
reaction-diffusion systems
singular perturbations and concentration phenomena
singularity formations in fluids

Relevant Degree Programs

Affiliations to Research Centres, Institutes & Clusters


Research Methodology

qualitative analysis


Master's students
Doctoral students
Any time / year round

De Giorgi conjectures, fractional equations, blow-ups in nonlinear parabolic and elliptic equations

I support public scholarship, e.g. through the Public Scholars Initiative, and am available to supervise students and Postdocs interested in collaborating with external partners as part of their research.
I am open to hosting Visiting International Research Students (non-degree, up to 12 months).

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Nov 2020)
New gluing methods and applications to nonlinear elliptic and parabolic equations (2020)

In this dissertation, we develop new gluing methods to construct concentration and blow-up solutions to some nonlinear elliptic and parabolic equations.In Chapter 2, we construct line bubbling solutions along boundary geodesics for the supercritical Lane-Emden-Fowler problem in low dimensions 6 and 7 by devising a new infinite dimensional reduction method.In Chapter 3, we construct type II finite time blow-up solutions to the energy critical heat equations in dimension 3, and the energy supercritical heat equation with cubic nonlinearity in dimensions 5, 6 and 7. The constructions rely on new inner-outer gluing method which aims at parabolic problems in low dimensions where slow decaying errors are present.In Chapter 4, by developing a new fractional gluing method, we construct infinite and finite blow-up solutions to the fractional heat equation with the critical exponent. In Chapter 5, we study the finite time singularity formation for the nematic liquid crystal flow in dimension two. We develop a new gluing method for this strongly coupled nonlinear system with non-variational structure and construct finite time blow-up solutions with precise profiles obtained.

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New solutions to local and non-local elliptic equations (2018)

We obtain a few existence results for elliptic equations.We develop in Chapter 2 a new infinite dimensional gluing scheme for fractionalelliptic equations in the mildly non-local setting. Here it is applied to the catenoid. As a consequence of this method, a counter-example to a fractional analogue of De Giorgi conjecture can be obtained [51].Then, in Chapter 3, we construct singular solutions to the fractional Yamabe problem using conformal geometry. Fractional order ordinary differential equations are studied.Finally, in Chapter 4, we obtain the existence to a suitably perturbed doublycritical Hardy–Schr¨odinger equation in a bounded domain in the hyperbolic space.

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Some new results on the SU(3) Toda system and Lin-Ni problem (2015)

In this thesis, we mainly consider two problems. First, we study the SU(3)Toda system. Let (M,g) be a compact Riemann surface with volume 1, h₁and h₂ be a C¹ positive function on M and p1; p2 ∈ ℝ⁺. The SU(3) Todasystem is the following one on the compact surface M[Formula and equation omitted]where ∆ is the Beltrami-Laplace operator, αq ≥ 0 for every q ∈ S₁, S₁ ⊂ M,Bq ≥ 0 for every q ∈ S₂,S₂ ⊂ M and q is the Dirac measure at q ∈ M. Weinitiate the program for computing the Leray-Schauder topological degreeof SU(3) Toda system and succeed in obtaining the degree formula for p1 ∈(0,4π)(4π,8π), p2 ∉ 4πℕ when S₁ = S₂ = 0.Second, we consider the following nonlinear elliptic Neumann problem{∆u-μu +uq =0 in Ω,u > 0 in Ω,au/av=0 on aΩ. where q=n+2/n-2, μ > 0 and Ω is a smooth and bounded domain in ℝn. Linand Ni (1986) conjectured that for μ small, all solutions are constants. Inthe second part of this thesis, we will show that this conjecture is false fora general domain in n = 4, 6 by constructing a nonconstant solution.

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Master's Student Supervision (2010 - 2018)
Effect of geometry on the behaviour of steady Newtonian fluid ina multiply connected domain (2014)

We start with the prototype problem of flow of a Newtonian fluid in the annular region between two infinitely long circular cylinders, with the givenvelocity and temperature on the boundaries of the domain.Then we will try to find out how does geometry affect the behavior of the flow inside of the domain. We will explore two invariant mappingsK_T and K_psi , such that under appropriate conditions on the boundary, the mapping K_T would preserve solution of temperature field from one domain to another and the mapping k_psi would preserve solution of velocity field.We will prove that if a mapping is conformal, it would preserve the convection-diffusion equation in both domains. After that, we will find which subsets of the conformals would also preserve the velocity field as well. In order to answer that question, we will obtain the required condition for the mapping , such that it would preserve both velocity and temperature fields, from one domain to another.

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