Titre :	Study of some non-classical boundary value problems for evolution equations
Titre original :	Étude de quelques problèmes aux limites non classiques pour des équations d’évolution
Type de document :	document multimédia
Auteurs :	Sofiane Benkouider, Auteur ; Abita Rahmoun, Directeur de thèse
Editeur :	Laghouat : Université Amar Telidji - Département de mathématiques
Année de publication :	2025
Importance :	133 p.
Accompagnement :	1 disque optique numérique (CD-ROM)
Note générale :	Option : Partial differential equations (PDEs)
Langues :	Anglais
Mots-clés :	Evolution heat equation  Parabolic equations  Positive initial energy  Blow- up time  Bounds for blow-up time  Sobolev spaces with variable exponents  Balakrishnan–Taylor damping  General decay rate  Time-varying delay  Convex function  Infinite memory
Résumé :	This thesis investigates the blow-up phenomena, asymptotic behavior, and stability of solutions for several classes of nonlinear partial differential equations (PDEs), including reaction-diffusion and wave-type equations with variable exponents, memory effects, and singular coefficients. The work is divided into four main parts. First, we study the blow-up phenomenon for nondegenerate parabolic PDEs in bounded domains. By considering a nonnegative diffusion coefficient a(x, t), we establish new blow- up criteria and derive sharp lower and upper bounds for the blow-up time of semilinear reaction-diffusion equations and nonlinear equations involving the m(x, t)-Laplacian op- erator. Second, we analyze the initial-boundary value problem for Kirchhoff-type viscoelastic wave equations with Balakrishnan–Taylor damping, infinite memory, and time-varying delay. Under suitable assumptions on the relaxation function and initial data, we prove that the energy decays at a rate determined by the relaxation function, which may be neither exponential nor polynomial. Moreover, we establish a general stability result under a weak growth condition on the relaxation kernel. Third, we examine a nonlinear viscoelastic wave equation with damping and source terms of variable-exponent type. We show that the energy grows exponentially, leading to finite-time blow-up in Lp(·)Γand Lk(·)Γnorms. For 2Γ≤ k(·)Γ< p(·), we obtain blow-up with positive initial energy and provide an upper bound for the blow-up time. When k(·)Γ=Γ2, we use the concavity method to prove blow-up and estimate its timing. For k(·)Γ≥ 2, we derive a lower bound for the blow-up time under certain conditions. Finally, we consider a p-Laplacian parabolic reaction-diffusion equation with a singular coefficient. We classify the results based on three initial energy levels: (i) sub-critical energy, where blow-up occurs with estimated bounds on the blow-up time; (ii) critical energy, where we prove both global existence (with asymptotic decay) and finite-time blow-up under different conditions, along with a lower bound for blow-up time; and (iii) super-critical energy, where solutions blow up in finite time with explicit bounds on the blow-up time. Our results generalize and refine existing literature, providing sharper blow-up es- timates, broader decay conditions, and new stability criteria for nonlinear PDEs with complex structural features.
note de thèses :	Thèse de doctorat en mathématiques

Study of some non-classical boundary value problems for evolution equations = Étude de quelques problèmes aux limites non classiques pour des équations d’évolution [document multimédia] / Sofiane Benkouider, Auteur ; Abita Rahmoun, Directeur de thèse . - Laghouat : Université Amar Telidji - Département de mathématiques, 2025 . - 133 p. + 1 disque optique numérique (CD-ROM).
Option : Partial differential equations (PDEs)
Langues : Anglais

Mots-clés :	Evolution heat equation  Parabolic equations  Positive initial energy  Blow- up time  Bounds for blow-up time  Sobolev spaces with variable exponents  Balakrishnan–Taylor damping  General decay rate  Time-varying delay  Convex function  Infinite memory
Résumé :	This thesis investigates the blow-up phenomena, asymptotic behavior, and stability of solutions for several classes of nonlinear partial differential equations (PDEs), including reaction-diffusion and wave-type equations with variable exponents, memory effects, and singular coefficients. The work is divided into four main parts. First, we study the blow-up phenomenon for nondegenerate parabolic PDEs in bounded domains. By considering a nonnegative diffusion coefficient a(x, t), we establish new blow- up criteria and derive sharp lower and upper bounds for the blow-up time of semilinear reaction-diffusion equations and nonlinear equations involving the m(x, t)-Laplacian op- erator. Second, we analyze the initial-boundary value problem for Kirchhoff-type viscoelastic wave equations with Balakrishnan–Taylor damping, infinite memory, and time-varying delay. Under suitable assumptions on the relaxation function and initial data, we prove that the energy decays at a rate determined by the relaxation function, which may be neither exponential nor polynomial. Moreover, we establish a general stability result under a weak growth condition on the relaxation kernel. Third, we examine a nonlinear viscoelastic wave equation with damping and source terms of variable-exponent type. We show that the energy grows exponentially, leading to finite-time blow-up in Lp(·)Γand Lk(·)Γnorms. For 2Γ≤ k(·)Γ< p(·), we obtain blow-up with positive initial energy and provide an upper bound for the blow-up time. When k(·)Γ=Γ2, we use the concavity method to prove blow-up and estimate its timing. For k(·)Γ≥ 2, we derive a lower bound for the blow-up time under certain conditions. Finally, we consider a p-Laplacian parabolic reaction-diffusion equation with a singular coefficient. We classify the results based on three initial energy levels: (i) sub-critical energy, where blow-up occurs with estimated bounds on the blow-up time; (ii) critical energy, where we prove both global existence (with asymptotic decay) and finite-time blow-up under different conditions, along with a lower bound for blow-up time; and (iii) super-critical energy, where solutions blow up in finite time with explicit bounds on the blow-up time. Our results generalize and refine existing literature, providing sharper blow-up es- timates, broader decay conditions, and new stability criteria for nonlinear PDEs with complex structural features.
note de thèses :	Thèse de doctorat en mathématiques