Topics in stability theory for partial differential equations.

摘要

Determining the stability of solutions is central to the analysis of partial differential equation (PDE) models arising in applications, as it is typically the stable solutions that are observed in practice. Topics related to the stability analysis of parabolic PDEs are discussed, including techniques used in determining the linear, nonlinear, and global stability of stationary solutions. For linear stability, the focus is both on determining the behavior of solutions when the spectrum of the linear operator is known, but lacks a spectral gap, and on locating the spectrum. Regarding the former, four examples are analyzed using renormalization groups, scaling variables, and spectral decompositions. In this analysis, a novel technique is applied that separates the solution into two components that naturally reflect the advection properties of the linear operator, allowing for the application of scaling variables and the creation of a spectral gap. To address the latter, a model of bioremediation, a process for cleaning contaminated soil, is considered. In this example, locating the spectrum is less straightforward. Geometric singular perturbation theory is employed to construct a traveling wave solution, and its properties are subsequently used in locating the spectrum of the associated linearized operator, thus determining the spectral stability of the wave. Nonlinear stability is then discussed. In general, when the linear operator lacks a spectral gap, the effects of the nonlinearity are not well understood. However, detailed information can be obtained in specific examples, three of which are presented. Existing results for the heat equation with polynomial nonlinearity are reviewed, as well as new results for nonlinear PDEs in which the linear operator is that which arises in the stability analysis of the traveling front in Burgers equation. Using the technique introduced in the linear stability analysis, invariant manifolds are constructed in the phase space of perturbations of this front. As a result, the asymptotic form of solutions will be determined, illustrating why their algebraic temporal decay rate can be increased by working in appropriate algebraically weighted Banach spaces. Finally, global stability is discussed, including the development of a Lyapunov functional argument for the traveling front in Burgers equation.