Parabolic systems of forward-backward type exhibiting (p,q)-type growth

Thesis

This thesis is concerned with systems of nonlinear equations exhibiting both forwardbackward type behaviour, and non-standard growth conditions. A motivating problem in one spatial dimension with application to the Met Office is discussed before proceeding to consider higher dimensional problems.

In the higher dimensional setting, in the absence of a monotonicity condition we work within the framework of Young measure solutions. We prove existence of large-data globalin- time Young measure solutions to initial-boundary-value problems for multidimensional nonlinear systems of forward-backward type of the form @_tu − div(a(Du)) + Bu = F, where B ∈ R^m×m, Bv·v ≤ 0 for all v ∈ R^m, F is an m-component vector-function defined on a bounded open Lipschitz domain Ω ⊂ Rⁿ, and a is a locally Lipschitz mapping of the form a(A) = K(A)A, where K : R^m×n → R. The long-time behaviour of these Young measure solutions is then studied, and under suitable assumptions on the source term we show convergence to Young measure solutions of the corresponding time-independent equations. We also discuss how the results proven can be adapted to cover mappings a which have different structure.

We develop a numerical algorithm for the approximate solution of problems in this class, and we prove the convergence of the algorithm to a Young measure solution of the system under consideration.

Authors: Caddick, M

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford