Well-posedness and asymptotic behavior for nonlinear wave equations with weighted nonlinear terms

Abstract

The wave equation is an important second-order linear partial dierential equation for the description of waves as they occur in physics such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetic, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange [11, 17, 64, 65]. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation [51]. Although many wave motion problems in physics can be modeled by the standard linear wave equation, or a similar formulation with a system of first-order equations, there are some exceptions. Perhaps the most important is water waves: these are modeled by the Laplace equation with time-dependent boundary conditions at the water surface (long water waves, however, can be approximated by a standard wave equation). Quantum mechanical waves constitute another example where the waves are governed by the Schrodinger equation and not a standard wave equation. Many wave phenomena also need to take nonlinear eects into account when the wave amplitude is significant