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 ? ×(0,+?) ? tt ? ? ? ? u=0 on ? ×(0,+?) 0 (1. 1) ? ? u+g(u)=0 on ? ×(0,+?) ? t 1 ? ? ? ? 0 1 u(x,0) = u (x); u (x,0) = 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? ×(0,+?) ? tt 0 ? ? ? ? u=0 on ? ×(0,+?) 0 (1. 2) ? ? u =?g(u )?f (u)on? ×(0,+?) ? t 1 1 ? ? ? ? 0 1 u(x,0) = u (x); u (x,0) = u (x),x? ? , t were widely studied in the literature, mainly when f =0,see[6,13,22]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=0,1, but assuming that f (s)s? 0, that is, f represents, for i i i each i, an attractive force.

State of the art in the fields of nonlinear analysis and nonlinear differential equations A tribute to the distinguished mathematician D.G. de Figueiredo Includes supplementary material: sn.pub/extras

Klappentext

This volume represents a broad survey of current research in the fields of nonlinear analysis and nonlinear differential equations. It is dedicated to Djairo G. de Figueiredo on the occasion of his 70 th birthday. The collection of 34 research and survey articles reflects the wide range of interests of Djairo de Figueiredo, including:

• various types of nonlinear partial differential equations and systems, in particular equations of elliptic, parabolic, hyperbolic and mixed type;
• equations of Schrödinger, Maxwell, Navier-Stokes, Bernoulli-Euler, Seiberg-Witten, Caffarelli-Kohn-Nirenberg;
• existence, uniqueness and multiplicity of solutions;
• critical Sobolev growth and connected phenomena;
• qualitative properties, regularity and shape of solutions;
• inequalities, a-priori estimates and asymptotic behavior;
• various applications to models such as asymptotic membranes, nonlinear plates and inhomogeneous fluids.
  The contributions of so many distinguished mathematicians to this volume document the importance and lasting influence of the mathematical research of Djairo de Figueiredo. The book thus is a source of new ideas and results and should appeal to graduate students and mathematicians interested in nonlinear problems.

Inhalt
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 Schrödinger 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 Schrödinger 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 Poincaré 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 forInhomogeneous 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.