# Juncheng Wei

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

Students must have taken the following courses:

1. partial differential equations

2. real analysis

3. functional analysis

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## Graduate Student Supervision

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Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

In this dissertation, we construct blow-up solutions for the critical heat equations and the two-dimensional Landau-Lifshitz-Gilbert equation.In Chapter 2, we construct a radial smooth positive ancient solution for the energy critical semi-linear heat equation in the Euclidean space with a dimension greater or equal to seven. It blows up at the origin with the profile of multiple Aubin-Talenti bubbles in the backward time infinity. In Chapter 3, we consider the Cauchy problem for the four-dimensional energy critical heat equation.We construct a positive infinite time blow-up solution with the blow-up rate ln t as t goes to infinity and study the stability of the infinite time blow-up solution. This gives rigorous proof of the infinite time blow-up predicted by Fila and King. In Chapter 4, we construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert (LLG) equation from the two-dimensional Euclidean space into the unit sphere with dimension two.Given any prescribed N points in the two-dimensional Euclidean space and a small positive constant T, we prove that there exists smooth initial data such that the solution blows up precisely at these points at finite time T, taking around each point the profile of sharply scaled degree 1 harmonic map.

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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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The present dissertation is split in three parts.The first considers the (unrestricted) Green tensor of Stokes system in the half-space. We derive the first ever pointwise estimates of such tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, and explore some applications of the pointwise estimates.The second part of this dissertation considers the magnetohydrodynamics equations (MHD equations) and the viscoelastic Navier–Stokes equations with damping. We construct self-similar and discretely self-similar solutions of both the MHD equations and the viscoelastic Navier–Stokes equations with damping with large initial data in the critical weak Lebesgue space.The third part of this dissertation deals with the Patlak–Keller–Segel–Navier–Stokes system. We prove the global existence of free-energy solutions with critical and subcritical mass.

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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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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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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

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

We show that the bounded solutions to the fractional Helmholtz equation, (−∆)ˢ u = u for 0

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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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