Decay of Solutions of a Degenerate Hyperbolic Equation

Abstract

This article studies the asymptotic behavior of solutions to the damped, non-linear wave equation ü + yů - m(||∇u||2)∆u = ƒ(x,t), for x ∈ Ω, t ≥ 0 u(x,0) = g(x), ů(x,0) = h(x), for x ∈ Ω u(x,t) = 0, for x ∈ ∂Ω t ≥ 0; where Ω is a bounded domain in Rn, with smooth boundary ∂Ω; y is a positive constant; m is a non-negative, bounded, and continuous function; ů denotes the derivative of u with respect to time; and as usual ∆u = n∑i=1 ∂2u/∂x2i, ||∇u||2 = n∑i=1 ƒΩ |∂u/∂xi|2 dx. This equation appears in mathematical physics as the Carrier or Kirchoff equation, when modeling planar vibrations. For a background and physical properties of this model, we refer the reader to [3], [4], [8], [12], and their references.