Decay of Solutions of a Degenerate Hyperbolic Equation
Date
1998-08-28
Authors
Dix, Julio G.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
This article studies the asymptotic behavior of solutions to the damped, non-linear wave equation
ü + yů - m(
∇u
<sup>2</sup>) ∆u = ƒ(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 nondegenerate 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
u̇
is square integrable on [0, ∞). We extend our results to systems and to related equations.
∇u
<sup>2</sup>) ∆u = ƒ(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 nondegenerate 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
u̇
is square integrable on [0, ∞). We extend our results to systems and to related equations.
Description
Keywords
Degenerate hyperbolic equation, Asymptotic behavior
Citation
Dix, J. G. (1998). Decay of solutions of a degenerate hyperbolic equation. <i>Electronic Journal of Differential Equations, 1998</i>(21), pp. 1-10.
Rights
Attribution 4.0 International
Rights Holder
This work is licensed under a Creative Commons Attribution 4.0 International License.