Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 21, pp. 1-10.
Title: Decay of solutions of a degenerate hyperbolic equation
Author: Julio G. Dix (Southwest Texas State Univ., San Marcos, Tx, USA)
Abstract:
This article studies the asymptotic behavior of solutions to
the damped, non-linear wave equation
$$
\ddot u +\gamma \dot u -m(\|\nabla u\|^2)\Delta u = f(x,t)\,,
$$
which is known as degenerate if the greatest lower bound for $m$
is zero, and non-degenerate if the greatest lower bound is positive.
For the non-degenerate case, it is already known that solutions decay
exponentially, but for the degenerate case exponential decay has remained
an open question. In an attempt to answer this question,
we show that in general solutions can not decay with exponential order,
but that $\|\dot u\|$ is square integrable on $[0, \infty)$.
We extend our results to systems and to related equations.
Submitted January 29, 1998. Published August 28, 1998.
Math Subject Classification: 35L05, 35B40.
Key Words: Degenerate hyperbolic equation; asymptotic behavior.