| Tóm tắt: | <p>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.</p> <p>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 @<sub>t</sub>u − div(a(Du)) + Bu = F, where B ∈ R<sup>m×m</sup>, Bv·v ≤ 0 for all v ∈ R<sup>m</sup>, F is an m-component vector-function defined on a bounded open Lipschitz domain Ω ⊂ R<sup>n</sup>, and a is a locally Lipschitz mapping of the form a(A) = K(A)A, where K : R<sup>m×n</sup> → 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.</p> <p>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.</p>
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