| Titre : | Contribution to the Qualitative Study of Some Generalized Boussinesq Problems |
| Auteurs : | Ines Garti, Auteur ; Mohamed Berbiche, Directeur de thèse |
| Type de document : | Thése doctorat |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2024 |
| Format : | 1 vol. (89 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Boussinesq equation, Cauchy problem, Stable set and Unstable set, Existence of global solution, Local existence, Finite time blow-up |
| Résumé : |
Research on Boussinesq equations has grown considerably over the past three decades due to their adaptability in explaining nonlinear phenomena, particularly in the analysis of water wave dynamics. These equations are widely used in various …elds, including the study of oceanic waves and coastal engineering. This thesis explores the existence and …nite-time blow-up of solutions in the Cauchy problem associated with a generalized Boussinesq equation. The study of the generalized Boussinesq equation and its solutions, including existence, non-existence, and blow-up, is of great interest. The thesis investigates solutions in both bounded and unbounded domains, as well as in the presence of logarithmic nonlinearity. Local solutions are proven to exist and be unique in both cases. Under certain restrictions on the initial data of our problems, we establish the existence and uniqueness of global solutions, as well as the possibility of blowing up solutions in …nite time. |
| Sommaire : |
Contents Dedication Abstract Résumé الملخص Acknowledgements Symbols and Abbreviations 1.1 Structure of Thesis . .. . . . 8 1 General Introduction 2 Preliminary Concepts 9 2.1 Functional spaces . . . . . . . . . . . . 9 2.1.1 Hilbert space . . . . . . . . . . . . . . . . 10 2.1.2 Banach space . . . . . . . . . . . . . . . . 10 2.1.3 Distributions spaces . . . . . . . 11 2.1.4 Lebesgue space . . . . . . . . . . . . . . . . . 13 2.1.5 Sobolev spaces . . . . . 14 vii2.1.6 Fourier transform . . . . . . . . . .. . . 16 2.2 Some inequalities . . . . . . . . . . . . . . . . . . . 18 2.3 Weak and Strong Convergence . . . . . . . .. . . 20 2.4 Abstract Cauchy problem . . . . . . . . . . 22 3 Initial Boundary Value Problem for the Dissipative Boussinesq Equation 24 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Existence and Nonexistence of Global solution . . . . . 25 3.3 Existence and Nonexistence of Global Solution for Boussinesq Equation With Logarithmic Nonlinearity (f(u) = u log juj) . . . . . . . . . . . . . . . 41 4 On the Cauchy Problem for the Generalized Double Dispersion Equation With Logarithmic Nonlinearity 48 4.1 Introduction . . . . . . . . . . . . . .. . . . 49 4.2 Mild solution . . . . . . . . . . . . . . . . . . 50 4.3 Linear estimates . . . . . . . . . . . . . . . . . . 58 4.4 Main resuts and their proofs . . . . . . . . . . 69 Conclusion 88 Bibliography 89 v |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/4992 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/158 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
