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Birkhauser Contributions To Nonlinear Analysis A Tribute To D.G. De Figueiredo On The Occasion Of His 70Th Birthday 2005 Edition by Thierry Cazenave David Costa Orlando Lopes
This paper is concerned with the existence and uniform decay rates of solutions of the waveequation with a sourceterm and subject to nonlinear boundary damping ? ? u ?? u =|u| u in ? x(0+?) ? tt ? ? ? ? u=0 on ? x(0+?) 0 (1. 1) ? ? u+g(u)=0 on ? x(0+?) ? t 1 ? ? ? ? 0 1 u(x0) = u (x); u (x0) = u (x)x? ? t n where ? is a bounded domain of R n? 1 with a smooth boundary ? = ? ?? . 0 1 Here ? and ? are closed and disjoint and ? represents the unit outward normal 0 1 to ?. Problems like (1. 1) more precisely ? u ?? u =?f (u)in? x(0+?) ? tt 0 ? ? ? ? u=0 on ? x(0+?) 0 (1. 2) ? ? u =?g(u )?f (u)on? x(0+?) ? t 1 1 ? ? ? ? 0 1 u(x0) = u (x); u (x0) = u (x)x? ? t were widely studied in the literature mainly when f =0see[61322]anda 1 long list of references therein. When f =0and f = 0 this kind of problem was 0 1 well studied by Lasiecka and Tataru [15] for a very general model of nonlinear functions f (s)i=01 but assuming that f (s)s? 0 that is f represents for i i i each i an attractive force. Table of contents : Remarks on a Class of Neumann Problems Involving Critical Exponents.- Existence of Solutions for a Class of Problems in IR N Involving the p(x)-Laplacian.- A Unitarian Approach to Classical Electrodynamics: The Semilinear Maxwell Equations.- Existence of Solutions for the Nonlinear Schroedinger Equation with V (?) = 0.- Asymptotic Behavior of a Bernoulli-Euler Type Equation with Nonlinear Localized Damping.- T-minima.- A Note on Heteroclinic Solutions of Mountain Pass Type for a Class of Nonlinear Elliptic PDE's.- Existence of Multiple Solutions for Quasilinear Equations via Fibering Method.- Symmetry of Solutions of a Semilinear Elliptic Problem in an Annulus.- Construction of a Radial Solution to a Superlinear Dirichlet Problem that Changes Sign Exactly Once.- Global Solvability and Asymptotic Stability for the Wave Equation with Nonlinear Boundary Damping and Source Term.- Multiscale Asymptotic Behavior of a Solution of the Heat Equation on .- Positive Solutions for a Class of Nonlocal Elliptic Problems.- On a Class of Critical Elliptic Equations of Caffarelli-Kohn-Nirenberg Type.- Existence and Number of Solutions for a Class of Semilinear Schroedinger Equations.- Multiparameter Elliptic Equations in Annular Domains.- Variational Principle for the Seiberg-Witten Equations.- Some Recent Results on Equations Involving the Pucci's Extremal Operators.- Principal Eigenvalue in an Unbounded Domain and a Weighted Poincare Inequality.- Uniform Stabilization for a Hyperbolic Equation with Acoustic Boundary Conditions in Simple Connected Domains.- Some Remarks on Semilinear Resonant Elliptic Problems.- Remarks on Regularity Theorems for Solutions to Elliptic Equations via the Ultracontractivity of the Heat Semigroup.- 2d Ladyzhenskaya-Solonnikov Problem for Inhomogeneous Fluids.- Generalization of a Well-known Inequality.- Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type.- On the Shape of Least-Energy Solutions to a Quasilinear Elliptic Equation Involving Critical Sobolev Exponents.- A-priori Bounds and Positive Solutions to a Class of Quasilinear Elliptic Equations.- The Role of the Equal-Area Condition in Internal and Superficial Layered Solutions to Some Nonlinear Boundary Value Elliptic Problems.- Some Recent Results Regarding Symmetry and Symmetry-breaking Properties of Optimal Composite Membranes.- Generic Simplicity for the Solutions of a Nonlinear Plate Equation.- An Estimate for the Blow-up Time in Terms of the Initial Data.- Lorentz Spaces and Nonlinear Elliptic Systems.- The Topology of Critical Sets of Some Ordinary Differential Operators.- A Note on the Superlinear Ambrosetti-Prodi Type Problem in a Ball.