The construction of blow-up solutions for some evolution equations
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Zhang, Qidi
issued: 2023-05-29T07:00:00Z
issued: 2023
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: 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.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/84705
note: The construction of blow-up solutions for some evolution equationsbyQidi ZhangA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)The University of British Columbia(Vancouver)May 2023© Qidi Zhang, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the thesis entitled:The construction of blow-up solutions for some evolution equationssubmitted by Qidi Zhang in partial fulfillment of the requirements for the degree of Doctor of Phi-losophy inMathematics.Examining Committee:Juncheng Wei, Professor, Department of Mathematics, UBCSupervisorStephen Gustafson, Professor, Department of Mathematics, UBCSupervisory Committee MemberTai-Peng Tsai, Professor, Department of Mathematics, UBCSupervisory Committee MemberMichael Jeffrey Ward, Professor, Department of Mathematics, UBCSupervisory Committee MemberYoung-Heon Kim, Professor, Department of Mathematics, UBCUniversity ExaminerJiahua Chen, Professor, Department of Statistics, UBCUniversity ExamineriiAbstractIn this dissertation, we construct blow-up solutions for the critical heat equations and the two-dimensionalLandau-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 atthe 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 equa-tion. We construct a positive infinite time blow-up solution with the blow-up rate ln t as t goes toinfinity and study the stability of the infinite time blow-up solution. This gives rigorous proof of theinfinite time blow-up predicted by Fila and King.In Chapter 4, we construct finite time blow-up solutions to…
has type: Thesis/Dissertation
date issued: 2023-11
is shown at: 10.14288/1.0432709
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Wei, Juncheng, 1968-
type: Text
identifier URI: http://hdl.handle.net/2429/84705