CONLEY INDEX CONTINUATION FOR SINGULARLY PERTURBED HYPERBOLIC EQUATIONS

in gratitude Abstract. Let Ω ⊂ R N, N ≤ 3, be a bounded domain with smooth boundary, γ ∈ L 2 (Ω) be arbitrary and φ: R → R be a C 1-function satisfying a subcritical growth condition. For every ε ∈]0, ∞ [ consider the semiflow πε on H 1 0 (Ω) × L2 (Ω) generated by the damped wave equation ε∂ttu + ∂tu = ∆u + φ(u) + γ(x) x ∈ Ω, t> 0, u(x, t) = 0 x ∈ ∂Ω, t> 0. Moreover, let π ′ be the semiflow on H1 0 (Ω) generated by the parabolic equation ∂tu = ∆u + φ(u) + γ(x) x ∈ Ω, t> 0, u(x, t) = 0 x ∈ ∂Ω, t> 0. Let Γ: H2 (Ω) → H1 0 (Ω) × L2 (Ω) be the imbedding u ↦ → (u, ∆u + φ(u) + γ). We prove in this paper that every compact isolated π ′-invariant set K ′ lies in H2 (Ω) and the imbedded set K0 = Γ(K ′) continues to a family Kε, ε ≥ 0 small, of isolated πε-invariant sets having the same Conley index as K ′. This family is upper-semicontinuous at ε = 0. Moreover, any (partially ordered) Morse-decomposition of K ′, imbedded into H1 0 (Ω) × L2 (Ω) via Γ, continues to a family of Morse decompositions of Kε, for ε ≥ 0 small. This family is again upper-semicontinuous at ε = 0. These results extend and refine some upper semicontinuity results for attractors obtained previously by Hale and Raugel

Annotated Type Systems for Program Analysis

Interpretation Table 1.2: Annotations in the Thesis In Chapter 2 we present a combined strictness and totality analysis.We are specifying the analysis as an annotated type system. The type system allows conjunctions of annotated types, but only at the top-level. The analysis is somewhat more powerful than the strictness analysis by Kuo and Mishra [KM89] due to the conjunctions and in that we also consider totality. The analysis is shown sound with respect to a natural-style operational semantics. The analysis is not immediately extendable to full conjunction. The analysis of Chapter 3 is also a combined strictness and totality analysis, however with "full" conjunction. Soundness of the analysis is shown with respect to a denotational semantics. The analysis is more powerful than the strictness analyses by Jensen [Jen92a] and Benton [Ben93] in that it in addition to strictness considers totality. So far we have only specified the analyses, however in order for the analyses to be practically useful we need an algorithm for inferring the annotated types. In Chapter 4 we construct an algorithm for the analysis of Chapter 2 The conjunctions are only allow at the "top-level". 1.3. OVERVIEW OF THESIS 25 3usingthelazy type approach by Hankin and Le Metayer [HM94a]. The reason for choosing the analysis from Chapter 3 is that the approach not applicable to the analysis from Chapter 2. In Chapter 5 we study a binding time analysis. We take the analysis specified by Nielson and Nielson [NN92] and we construct an more e#cient algorithm than the one proposed in [NN92]. The algorithm collects constraints in a structural manner as the algorithm T [Dam85]. Afterwards the minimal solution to the set of constraints is found. The analysis in Chapter 6 is specified by abstract interp..

Your memory will always be with me

Without your love and support this would not have been possible iii Acknowledgements The completion of this dissertation was only possible with the support and contributions of many people. I would like to begin by thanking my husband, Blaine, for standing behind me and for believing that I would finish this dissertation. Without his love, support and understanding, this dissertation would not be. I would like to thank my daughter Joei, for not feeling neglected when I did not have time to do those “girly ” things with her. I would also like to thank my mother, Jessie H. Judson, for instilling in me the importance of a good education and making me believe that the world was mine to conquer. Behind the scenes were my friends and family who were always there for me during my doctoral studies. I am beholden to you all for your fellowship, reassurance, and support. I would like to thank the family who gave the gift of life on February 7, 1998. Without their sacrifice and thoughtfulness, I would not have been able to enjoy the things that life has to bring nor complete this research. I thank you from the bottom of my heart. Special thanks to Dr. Doris L. Carver, my advisor, for her generous support, guidance, and patience an