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.
Spis treści
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 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.
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