Partial Differential Equations and Applications: Colloquium in Honor of Hamidou Touré, Ouagadougou, Burkina Faso, November 5–9, 2018 (Springer Proceedings in Mathematics & Statistics, 420)

Preface
	Préface (Version française)
	Préface (Version anglaise)
	Avant Propos (Version française)
	Avant Propos (Version anglaise)
Contents
Contributors
1 Existence and Uniqueness of Solution for Semi-linear Conservation Laws with Velocity Field in L∞
	1.1 Introduction
	1.2 Position of the Problem
		1.2.1 Statement of the Aim and Functional Setting
		1.2.2 Space–Time Least-Square and Linear Problem
		1.2.3 Space–Time Least-Square and Semi-linear Problem
	1.3 Existence and Qualitative Results
		1.3.1 Existence and Uniqueness
	1.4 Existence and Uniqueness Result for the Penalization Version
	1.5 Numerical Study and Simulations
		1.5.1 A Finite-Element Method for Semi-linear Conservations Laws
			1.5.1.1 A Finite-Element Method and Picard\'s Iteration
			1.5.1.2 A Finite-Element Method and Newton\'s Method
		1.5.2 STILS for Semi-linear Conservations Laws
			1.5.2.1 STILS and Picard\'s Iteration
		1.5.3 STILS Adaptive Newton Method
		1.5.4 Numerical Experiment
	References
2 Structural Stability of p(x)-Laplace Problems with Robin-Type Boundary Condition
	2.1 Introduction
	2.2 Preliminaries
	2.3 Renormalized Solution
		2.3.1 Proof of Theorem 2.3.1
			2.3.1.1 Approximate Solutions
			2.3.1.2 Convergence Results
			2.3.1.3 Existence of Renormalized Solution
		2.3.2 Uniqueness of Renormalized Solution
	2.4 Continuous Dependence of Renormalized Solution
	References
3 Weak Solutions of Anti-periodic Discrete Nonlinear Problems
	3.1 Introduction
	3.2 Mathematical Background
	3.3 Existence of Weak Solutions
	3.4 An Extension
	References
4 Boundary Feedback Controller over a Bluff Body for Prescribed Drag and Lift Coefficients
	4.1 Introduction
	4.2 Notation and Preliminaries
		4.2.1 Function Spaces
		4.2.2 Linear Forms and a Few Inequalities
	4.3 Existence Result
		4.3.1 The Variational Formulation
		4.3.2 A Priori Estimates
		4.3.3  A Galerkin Basis for the Space W(Ω)
		4.3.4 Existence of Weak Solution
			4.3.4.1 The Galerkin Method
		4.3.5 Retrieving the Controlled Problem
	4.4 Concluding Remarks
	References
5 Discrete Potential Boundary-Value Problems of Kirchhoff Type
	5.1 Introduction
	5.2 Preliminary
	5.3 Proof of the Existence of Classical Solutions by Ground State Method
	5.4 Proof of the Existence of Classical Solutions by Mountain Pass Method
	References
6 From Calculus of Variation to Exterior Differential Calculus: A Presentation and Some New Results
	6.1 Introduction
	6.2 Exterior Differential Calculus
		6.2.1 Linear and Differential Forms
		6.2.2 Exterior k-form
		6.2.3 Exterior Product
		6.2.4 Differential Forms and Exterior Differentiation
		6.2.5 Pullback
		6.2.6 Poincaré Theorem
		6.2.7 Darboux Theorem
	6.3 Exterior Differential System
		6.3.1 Introductive Examples
		6.3.2 The General Problem
		6.3.3 The Regularity Condition
		6.3.4 The Main Theorem
		6.3.5 An Example
	6.4 Main Result: Douglas Problem
	References
7 Existence of Local and Maximal Mild Solutions for Some Non-autonomous Functional Differential Equation with Finite Delay
	7.1 Introduction
	7.2 Preliminary
	7.3 Existence of Local and Maximal Mild Solutions
	7.4 Application
	References
8 Existence, Regularity, and Stability in the α-Norm for Some Neutral Partial Functional Differential Equations in Fading Memory Spaces
	8.1 Introduction
	8.2 Analytic Semigroup, Fractional Power of Its Generator, and Partial Functional Differential Equations
	8.3 Existence and Uniqueness of Strict Solutions
	8.4 Smoothness Results of the Operator Solution
	8.5 Linearized Stability of Solutions
	8.6 Application
	References
9 Pseudo-almost Periodic Solutions of Class r in the α-Norm Under the Light of Measure Theory
	9.1 Introduction
	9.2 Analytic Semigroup
	9.3 Spectral Decomposition
	9.4 (μ,ν)-Pseudo-almost Periodic Functions
	9.5 Properties of (μ,ν)-Pseudo-almost Periodic Functions of Class r
	9.6 (μ,ν)-Pseudo-almost Periodic Solutions of Class r
	9.7 Application
	References
10 Global Stability for a Delay SIR Epidemic Model with General Incidence Function, Observers Design
	10.1 Introduction
	10.2 Model Presentation
	10.3 Basic Reproduction Number and Equilibria
	10.4 Stability of Equilibria
		10.4.1 Stability of the Disease-Free Equilibrium
		10.4.2 Global Stability of the Endemic Equilibrium
	10.5 Observer Design
		10.5.1 A Simple Observer for an SIR Epidemic Model
		10.5.2 High-Gain Observer for an SIR Epidemic Model
	10.6 Numerical Simulation and Comments
	10.7 Conclusion
	References
11 Threshold Parameters of Stochastic SIR and SIRS Epidemic Models with Delay and Nonlinear Incidence
	11.1 Introduction
	11.2 Extinction of the Epidemic Model (11.1)
	11.3 Persistence in Mean of the Epidemic Model (11.1)
	11.4 The Threshold of the Stochastic SIRS Epidemic Model
	11.5 Conclusion
	References
12 Weak Solutions for Nonlinear Boltzmann–Poisson System Modelling Electron–Electron Interactions
	12.1 Introduction
		12.1.1 Assumptions and Notations
	12.2 Properties of the Dynamics
		12.2.1 Continuity of Q0
		12.2.2 Entropy Inequalities and Invariants of Collisions
	12.3 Free-Space Vlasov Equation
		12.3.1 Renormalized Solution
		12.3.2 Penalization Method
	12.4 Equation of Boltzmann with Specular Reflection Boundary Condition
		12.4.1 Existence of a Weak Solution
		12.4.2 Energy Estimate
	12.5 Stability Results
		12.5.1 L1-weak Precompacity Q1(Fα)
		12.5.2 Velocity-Averaging Lemma
		12.5.3 L1-compactness of L5(fn)
		12.5.4 Compactness of L1(fn,fn)
		12.5.5  L1-Compactness of L1(fn,fn)
		12.5.6 Analysis of L2, L3, and L4
	12.6 The Boltzmann–Poisson System
		12.6.1 Main Result
		12.6.2 The Modified Boltzmann–Poisson System
		12.6.3 Unmodified Boltzmann–Poisson System
	References