Evolutionary Dynamic Equations
Description
The book discusses the stability, observability, and controllability of nonlinear systems of PDEs (such as Wave, Heat, Euler-Bernoulli beam, Petrovsky, Kirchhoff, equations, and more). Methods based on the theory of classical weak functions analysis and movements in Sobolev spaces are used to analyze nonlinear systems of evolutionary partial differential equations. With the unifying theme of evolutionary dynamic equations, both linear and nonlinear, in more complex environments with different approaches, the book presents a multidisciplinary blend of topics, spanning the fields of PDEs applied to various models coming from theoretical physics, biology, engineering, and natural sciences.
This comprehensive book is prepared for a diverse audience interested in applied mathematics. With its broad applicability, this book aims to foster interdisciplinary collaboration and facilitate a deeper understanding of complex phenomenon concepts, practically in electromagnetic waves, the acoustic model for seismic waves, waves in blood vessels, wind drag on space, the linear shallow water equations, sound waves in liquids and gases, non-elastic effects in the string.
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Persons
Akram Ben Aissa was born on June 26, 1986, in Eljem, Tunisia. A graduate with high honors from the Faculty of Sciences in Monastir, Akram's academic prowess earned him a Master's degree in Mathematics in 2011. In 2016, Akram earned his Ph.D. in Mathematics from the University of Monastir. His thesis, "Grushin problems and Control theory of PDEs". Akram's research interests span Control theory of Partial Differential Equations, Functional Analysis, and Stability and Control of PDEs. With an impressive publication record, including works on viscoelastic wave equations and second-order evolution equations, he continues to shape the mathematical landscape. In 2022, armed with a Habilitation à diriger des recherches (HdR) from the University of Sousse, he is now an associate Professor at University of Sousse, Tunisia.
Khaled Zennir was born in Algeria 1982. He received his PhD in Mathematics in 2013 from Sidi Bel Abbès University, Algeria (Assist. professor). He is now associate Professor at Qassim University, KSA. His research interests lie in Nonlinear Hyperbolic Partial Differential Equations: Global Existence, Blow-Up, and Long Time Behavior.
Content
- Intro
- Preface
- Contents
- 1 Introduction
- 1.1 Basic knowledge
- 1.1.1 Necessary contractions in the C0-semigroup approach
- 1.1.2 Useful inequalities
- 2 Qualitative properties for impulsive wave equation: controllability and observability
- 2.1 Classical solution for the first-order impulsive wave equation
- 2.2 Impulse controllability
- 2.3 Impulse observability inequality
- 3 Viscoelastic wave equation with dynamic boundary conditions
- 3.1 Well-posedness of the problem via Lumer-Phillips theorem
- 3.2 C0-semigroup approach and stability
- 4 Passage from internal exact controllability of beam equation to pointwise exact controllability
- 4.1 Estimation and regularity results near a point
- 4.2 Internal exact controllability of the beams equation
- 4.3 An inverse inequality
- 4.4 Estimates on the controls
- 4.5 The weak* convergence
- 5 Second-order evolution equations with/without delay
- 5.1 Relation between the decay rate of the energy for systems with/without delay
- 5.2 The exponentially and polynomial rate
- 5.3 Applications for models with interior damping
- 5.3.1 Wave equation
- 5.3.2 The multidimensional wave equation
- 5.3.2.1 The influence of internal damping
- 5.3.2.2 The impact of boundary damping
- 5.3.3 Weakly coupled and partially damped with boundary delay
- 5.3.4 Euler-Bernoulli beam
- 6 Euler-Bernoulli beam conveying fluid equation with nonconstant velocity and dynamical boundary conditions
- 6.1 Well-posedness via Lax-Milgram theorem
- 6.2 The influence of the density and velocity on stability
- 7 Stabilization of dissipative nonlinear evolution models
- 7.1 Nonlinear Petrovsky-wave system
- 7.1.1 Faedo-Galerkin approach
- 7.1.2 Stability via multiplier method
- 7.2 Nondegenerate Kirchhoff system coupled with heat conduction
- 7.2.1 Unique weak solution
- 7.2.2 Exponential stability
- 7.3 Nondegenerate Kirchhoff equation with localized nonlinear damping
- 8 Nonlinear Petrovsky-type models
- 8.1 Delayed Petrovsky equation with a nonlinear strong damping
- 8.1.1 Global solvability in Sobolev spaces
- 8.1.2 Stability for the energy
- 8.2 Petrovsky equation with a nonlinear strong dissipation
- 8.2.1 Well-posedeness and regularity
- 8.2.2 Asymptotic behavior via Lyapunov functional
- 8.3 Examples
- Bibliography
- Index
