- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- New gluing methods and applications to nonlinear elliptic...
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
New gluing methods and applications to nonlinear elliptic and parabolic equations Zhou, Yifu
Abstract
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.
Item Metadata
Title |
New gluing methods and applications to nonlinear elliptic and parabolic equations
|
Creator | |
Publisher |
University of British Columbia
|
Date Issued |
2020
|
Description |
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.
|
Genre | |
Type | |
Language |
eng
|
Date Available |
2020-07-27
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0392557
|
URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
|
Graduation Date |
2020-11
|
Campus | |
Scholarly Level |
Graduate
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International