| Titre : | Qualitative study of certain evolution problems |
| Auteurs : | Messaouda Terchi, Auteur ; Mohamed Berbiche, Directeur de thèse |
| Type de document : | Monographie imprimée |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2020 |
| Format : | 1 vol. (92 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Damped wave equation,Heat system,damped wave system,Local existence,Global existence,Asymptotic behavior,Finite time blow-up |
| Résumé : |
In this thesis, we are interested in the study of the existence and uniqueness of globalsolutions, as well as, the blow up in …nite time of solutions for a certains systems of semilinear Volterra integro di¤erential equations of parabolic and hyperbolic type. Especially the non-linear part is de…ned by an integral terms over the past history of the nonlinear forcing containing fractional time-dependent convolution kernels. We study this type of generalized problems to obtain similar results to those obtained in the case of an equation. We will see that under certain conditions on the exponents, the order of the temporal
fractional derivatives there is a critical value of the dimension space for which the global with small data solution results as well as the explosion in …nite time with initial conditions having positive average are obtained. The methodology to be followed to demonstrate the global existence and the asymptotic behavior based essentially on the use of the semi-group method combined with a priori estimates in the Lebesgue spaces. In parallel, in the study of the blow-up in …nite time result, we will focus on the concept of weak solutions and its connection with the mild ones and thus via the test functions method’s get the desired results. |
| Sommaire : |
Dédicace i
Abstract i Acknowledgements ii Symbols and Abbreviations iii Table Of Contents iv 1 General Introduction 2 1.1 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Presentation of the Obtained Results . . . . . . . . . . . . . . . . . . . . . 8 2 Preliminary Concepts 13 2.1 De…nitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Fractional integrals and derivatives . . . . . . . . . . . . . . . . . . 15 2.2 Semigroup of bounded operators . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 m-dissipative operators . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 The Laplacian in an open subset of RN : L2; C0 theories . . . . . . . 18 2.3 Contraction semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Heat semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Damped Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 v2.4.1 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Blow-up Of Solution For Damped Wave System With Nonlinear Memory 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Blow-up Theorem and its Proof . . . . . . . . . . . . . . . . . . . . . . . . 28 4 On The Nonexistence of Global Solution For Wave Equations With Double Damping Terms and Nonlinear Memory 38 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Some preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Local-existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4 Blow-up results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5 Global Small Data Solution For a System of Semilinear Heat Equations and The Corresponding System of Damped Wave Equations With Nonlinear Memory 57 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4 Proof of global existence theorem of the heat system . . . . . . . . . . . . 73 5.5 Damped wave system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5.1 Proof of theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Conclusion 92 Bibliography 92 |
| En ligne : | http://thesis.univ-biskra.dz/4992/1/These%20Terchi%20Messaouda.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/102 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
