- Title Information
- Title
- Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs
- Name:
Personal
- Name Part
- Ben-Gal, Nitsan
- Role
- Role Term:
Text
- creator
- Origin Information
- Copyright Date
(keyDate="yes", encoding="w3cdtf")
- 2009
- Physical Description
- Extent
- xvi, 187 p.
- digitalOrigin
- born digital
- Note
- Thesis (Ph.D.) -- Brown University (2009)
- Name:
Personal
- Name Part
- Fieldler, Bernold
- Role
- Role Term:
Text
- director
- Name:
Personal
- Name Part
- Dafermos, Constantine
- Role
- Role Term:
Text
- reader
- Name:
Personal
- Name Part
- Sandstede, Bjorn
- Role
- Role Term:
Text
- reader
- Name:
Corporate
- Name Part
- Brown University. Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Genre (aat)
- theses
- Abstract
- In recent years, there has been a great deal of interest surrounding the study of the asymptotics and global attractor structure for scalar parabolic PDEs which are either
dissipative or undergo finite-time blow-up. This thesis presents solutions to the asymptotics and connection problems for slowly non-dissipative scalar PDEs, i.e. the final remaining class of
scalar parabolic reaction-diffusion equations. Such PDEs produce solutions that neither blow up nor are dissipative. These "grow-up" solutions grow to infinite norm in infinite time, and it is
the added challenges they introduce that are overcome in this thesis. In the pursuit of this result, a number of new concepts are defined, including slowly non-dissipative PDEs, non-compact
global attractors, and the completed inertial manifold. Many of the underlying assumptions used in the study of dissipative PDEs are inapplicable to slowly non-dissipative PDEs. Thus, concepts
must be redefined and techniques updated or extended. The effects of a slowly non-dissipative nonlinearity on the global bifurcation diagram are investigated, and the y-map, a technique critical
to solving the connection problem for dissipative systems, is extended to slowly non-dissipative PDEs and a wider range of boundary conditions. The completed inertial manifold is introduced and
proven to exist for certain classes of slowly non-dissipative equations. The development of this new structure and its advantageous characteristics provide the tools necessary to prove
convergence for grow-up solutions which form heteroclinic connections to infinity. The asymptotics of grow-up solutions are determined, and via combining the various expanded techniques, a full
decomposition of the non-compact global attractor is produced for a generic choice of nonlinearity. The results are studied for a selection of interesting cases and are shown to hold for both
Neumann and Dirichlet boundary conditions.
- Subject (Local)
- Topic
- grow-up
- Subject (Local)
- Topic
- global attractor
- Subject (Local)
- Topic
- semilinear parabolic PDE
- Subject (Local)
- Topic
- asymptotic behavior
- Subject (Local)
- Topic
- connecting orbits
- Subject (Local)
- Topic
- inertial manifold
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20091218
- Language
- Language Term:
Code (ISO639-2B)
- eng
- Language Term:
Text
- English
- Identifier:
DOI
- 10.7301/Z0MW2FFK
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Type of Resource (primo)
- dissertations