DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

QUENCHING OF SOLUTIONS OF NONLINEAR HYPERBOLIC
EQUATIONS WITH DAMPING
JIANMIN ZHU
A hyperbolic initial-boundary value problem with nonlinear damping and singular
source terms is studied. A criterion for a solution to reach the value 1 in a finite time
is established.
Copyright © 2006 Jianmin Zhu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The concept of quenching was introduced in 1975 by Kawarada [6] through a first initialboundary
value problem for a semilinear heat equation. Chang and Levine [4]extended
the concept to a first initial-boundary value problem for a semilinear wave equation in
1981. Over more than twenty years, there has been an extensive study on quenching of
solution to various partial differential equations, particularly for parabolic equations. Because
of the lack of a maximum principle for hyperbolic equations as useful as that for
parabolic equations, quenching phenomena for hyperbolic equations have not been studied
as extensively as for parabolic equations. The study of quenching phenomena for hyperbolic
initial-boundary value problems has been focused on two types of problems:
one with singular nonlinearities in the differential equations, and the other with singular
nonlinearities in the boundary conditions. Chang and Levine [4] considered the quenching
problem to a first initial-boundary value problem for a semilinear wave equation with
singular nonlinearities in the differential equations in 1981. Later, Smith [12] and Levine
and Smiley [9] generalized the results to the multidimensional case. The effect of nonlinear
boundary conditions on the homogeneous wave equation was investigated by Levine
[7] in 1-dimensional space while Rammaha [10] in the multidimensional space. For other
related works, we refer the reader to [2, 8, 11] and the references therein.
For the initial-boundary value problem with nonlinear damping and source terms,
Georgiev and Todorova [5] studied the existence and the blow-up of solutions in 1994,
Chan and Zhu [3] studied the corresponding quenching problem, and Agre and
Rammaha [1] studied the existence and the quenching of solutions for a wave equation
in one space dimension. Here, we would like to study the quenching phenomena for the
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1187–1194