In this edited volume leaders in the field of partial differential equations present recent work on topics in PDEs arising from geometry and physics. The papers originate from a 2015 research school organized by CIMPA and MIMS in Hammamet, Tunisia to celebrate the 60th birthday of the late Professor Abbas Bahri. The opening chapter commemorates his life and work. While the research presented in this book is cutting-edge, the treatment throughout is at a level accessible to graduate students. It includes short courses offering readers a unique opportunity to learn the state of the art in evolution equations and mathematical models in physics, which will serve as an introduction for students and a useful reference for established researchers. Finally, the volume includes many open problems to inspire the next generation.

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Cover; Series page; Title page; Copyright information; Table of contents; Preface; Abbas Bahri: A Dedicated Life; 0.1 A short biography; 0.2 Mathematical contributions; References; 1 Blow-up Rate for a Semilinear Wave Equation with Exponential Nonlinearity in One Space Dimension; 1.1 Introduction; 1.2 The Local Cauchy Problem; 1.3 Energy Estimates; 1.4 ODE Type Estimates; 1.4.1 Preliminaries; 1.4.2 The Blow-up Rate; 1.5 Blow-up Estimates for Equation (1.1); 1.5.1 Blow-up Estimates in the General Case; 1.5.2 Blow-up Estimates in the Non-characteristic Case; References

2 On the Role of Anisotropy in the Weak Stability of the Navier-Stokes System2.1 Introduction and Statement of Results; 2.1.1 Setting of the Problem; 2.1.2 Statement of the Main Result; 2.1.3 Layout; 2.2 Proof of the Main Theorem; 2.2.1 General Scheme of the Proof; 2.2.2 Anisotropic Profile Decomposition; 2.2.3 Propagation of Profiles; 2.2.4 End of the Proof of the Main Theorem; 2.3 Profile Decomposition of the Sequence of Initial Data: Proof of Theorem 2.12; 2.3.1 Profile Decomposition of Anisotropically Oscillating, Divergence-free Vector Fields

2.3.2 Regrouping of Profiles According to Horizontal Scales2.4 Proof of Theorems 2.14 and 2.15; 2.4.1 Proof of Theorem 2.14; 2.4.2 Proof of [interactionprofilescale1]Theorem 2.15; References; 3 The Motion Law of Fronts for Scalar Reaction-diffusion Equations with Multiple Wells: the Degenerate Case; 3.1 Introduction; 3.1.1 Main Results: Fronts and Their Speed; 3.1.2 Regularized Fronts; 3.1.3 Paving the Way to the Motion Law; 3.1.4 A First Compactness Result; 3.1.5 Refined Estimates Off the Front Set and the Motion Law; 3.2 Remarks on Stationary Solutions

3.2.1 Stationary Solutions on R with Vanishing Discrepancy3.2.2 On the Energy of Chains of Stationary Solutions; 3.2.3 Study of the Perturbed Stationary Equation; 3.3 Regularized Fronts; 3.3.1 Finding Regularized Fronts; 3.3.2 Local Dissipation; 3.3.3 Quantization of the Energy; 3.3.4 Propagating Regularized Fronts; 3.4 First Compactness Results for the Front Points; 3.5 Refined Asymptotics Off the Front Set; 3.5.1 Relaxations Towards Stationary Solutions; 3.5.2 Preliminary Results; 3.5.3 The Attractive Case; 3.5.4 The Repulsive Case; 3.5.5 Estimating the Discrepancy; Linear Estimates

Includes bibliographical references.

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