Juncheng Wei

Professor

Research Classification

Research Interests

Mathematics
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

Recruitment

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 - May 2021)
An analysis of localized patterns in some novel reaction diffusion models (2021)

In this thesis we investigate strongly localized solutions to systems of singularly perturbed reaction-diffusion equations arising in several new contexts. The first such context is that of bulk-membrane-coupled reaction diffusion systems in which reaction-diffusion systems posed on the boundary and interior of a domain are coupled. In particular we analyze the consequences of introducing bulk-membrane-coupling on the behaviour of strongly localized solutions to the singularly perturbed Gierer-Meinhardt model posed on the one-dimensional boundary of a flat disk and the singularly perturbed Brusselator model posed on the two-dimensional unit sphere. Using formal asymptotic methods we derive hybrid numerical-asymptotic equations governing the structure, linear stability, and slow dynamics of strongly localized solutions consisting of multiple spikes. By numerically calculating stability thresholds we illustrate that bulk-membrane coupling can lead to both the stabilization and the destabilization of strongly localized solutions based on intricate relationships between the bulk-membrane-coupling parameters. The remainder of the thesis focuses exclusively on the singularly perturbed Gierer-Meinhardt model in two new contexts. First, the introduction of an inhomogeneous activator boundary flux to the classically studied one-dimensional Gierer-Meinhardt model is considered. Using the method of matched asymptotic expansions we determine the emergence of \textit{shifted} boundary-bound spikes. By linearizing about such a shifted boundary-spike solution we derive a class of \textit{shifted} nonlocal eigenvalue problems parametrized by a shift parameter. We rigorously prove partial stability results and by considering explicit examples we illustrate novel phenomena introduced by the inhomogeneous boundary fluxed. In the second and final context we consider the Gierer-Meinhardt model in three-dimensions for which we use formal asymptotic methods to study the structure, stability, and dynamics of strongly localized solutions. Most importantly we determine two distinguished parameter regimes in which strongly localized solutions exist. This is in contrast to previous studies of strongly localized solutions in three-dimensions where such solutions are found to exist in only one parameter regime. We trace this distinction back to the far-field behaviour of certain core problems and formulate an appropriate conjecture whose resolution will be key in the rigorous study of strongly localized solutions in three-dimensional domains.

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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 - 2020)
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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