Abstract

Van Schaftinen showed that the inequalities of Bourgain and Brezis give rise to new function spaces that refine the classical embedding of critical Sobolev spaces into BMO. It was suggested by Van Schaftingen that similar results should hold in the setting of bounded domains.

The first part of this thesis contains the proofs of this conjecture as well as the development of a non-homogeneous theory of Van Schaftingen spaces. Based on the results in the non-homogeneous setting, we are able to show that the refined embeddings can also be established for bmo spaces on Riemannian manifolds with bounded geometry, introduced by Taylor.

The stability of parabolic equations with time delay plays an important role in the study of non-linear reaction-diffusion equations with time delay. While the stability regions for such equations without convection on bounded time intervals were described by Travis and Webb, the problem remained unaddressed for the equations with convection. The need to determine exact regions of stability for such equations appeared in the context of the work of Mei and Wang on the Nicholson equation with delay.

In the second part of this thesis, we study the parabolic equations with and without convection. It has been shown that the presence of convection terms can change the regions of stability. The implications for the stability problems for non-linear equations are also discussed.