Electron. J. Diff. Eqns., Vol. 1998(1998), No. 21, pp. 1-10.

Decay of solutions of a degenerate hyperbolic equation

Julio G. Dix

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. August, 28, 1998.
Math Subject Classification: 35L05, 35B40.
Key Words: Degenerate hyperbolic equation, asymptotic behavior.

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Julio G. Dix
Department of Mathematics
Texas State University
San Marcos, TX 78666 USA
e-mail: jd01@swt.edu

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