| Titre : | Qualitative study of some viscoelastic evolution problems |
| Auteurs : | Ammar Melik, 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, 2023 |
| Format : | 1 vol. (98 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | parabolic equation, viscoelasticity, critical Fujita exponent, global existence, energy estimate, decay estimates, exponential stability. |
| Résumé : |
In this thesis, we consider the cauchy problem for weakly coupled systems of fractional semilinear Volterra integro di?erential equations of pseudo-parabolic type with a memory term in multi-dimensional space Rn (n ? 1), under small initial data and the conditions on the convolution kernel k which are weaker than the classical di?erential inequalities, we establish new results for exponential decay of solutions for single equation of the systems in the Fourier space, and we prove the global existence and uniqueness of solutions for weakly coupled systems where data are supposed to belong to di?erent classes of regularity by introducing a set of time-weighted Sobolev spaces and applying the contracting mapping theorem. |
| Sommaire : |
Abstract i Acknowledgements iv Symbols and Abbreviations v 1 Preliminary Concepts 9 1.1 Functional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Spaces of test functions and distributions . . . . . . . . . . . . . . . 10 1.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 The Space S (Rn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Fourier transform of tempered distributions . . . . . . . . . . . . . 15 1.4 Some important inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.1 Classical Gagliardo-Nirenberg inequality . . . . . . . . . . . . . . . 17 1.4.2 Fractional Gagliardo-Nirenberg inequality . . . . . . . . . . . . . . 18 1.4.3 Fractional Leibniz rule . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.4 Fractional chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.5 Fractional powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.6 Gronwall's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.7 Banach Contraction-Mapping Principle . . . . . . . . . . . . . . . . 20 2 Global existence and decay estimates for the semilinear heat equation with memory in Rn. 21 2.1 Mild solution formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Decay properties of solution operators . . . . . . . . . . . . . . . . . . . . . 25 2.3 Decay estimates for linear problem . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Global existence and decay estimates for semi-linear problem . . . . . . . . 35 3 Global existence and decay estimates for the semilinear nonclassical- di?usion equations with memory in Rn. 43 3.1 Mild solution formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Decay Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Decay estimates for linear problem . . . . . . . . . . . . . . . . . . . . . . 59 3.4 Global existence and decay for semi-linear problem . . . . . . . . . . . . . 62 4 Global existence and decay estimates of solutions for a system of semi- linear heat equations with memory involving the fractional Laplacian. 75 4.0.1 Linear cauchy problem with memory-type dissipation . . . . . . . . 76 4.1 Semi-linear Cauchy problem with memory-type dissipation . . . . . . . . . 77 4.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1.3 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Global existence and decay estimates of solutions for a weakly coupled system of semi-linear cauchy problem with memory-type dissipation . . . . . . 88 4.2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2.2 Initial data from Sobolev spaces . . . . . . . . . . . . . . . . . . . . 90 4.2.3 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2.4 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Conclusion 98 Bibliography 98 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6333 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/154 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
