Description
- 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.
- Notes:
- Thesis (Ph.D.) -- Brown University (2009)
Access Conditions
- Rights
- In Copyright
- Restrictions on Use
- Collection is open for research.
Citation
Ben-Gal, Nitsan,
"Grow-Up Solutions and Heteroclinics to Infinity for Scalar Parabolic PDEs"
(2009).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0MW2FFK
Relations
Collection:
-
Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....