International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 17, Pages 1073-1081
doi:10.1155/S0161171203204361
Time estimates for the Cauchy problem for a third-order hyperbolic equation
Department of Mathematics, University of Texas - Pan American, Edinburg 78539-2999, TX, USA
Received 4 April 2002
Copyright © 2003 Vladimir Varlamov. 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.
Abstract
A classical solution is considered for the Cauchy problem:
(utt−Δu)t+utt−αΔu=f(x,t), x∈ℝ3, t>0; u(x,0)=f0(x), ut(x,0)=f1(x), and
utt(x)=f2(x), x∈ℝ3, where α=const,
0<α<1. The above equation governs the propagation of
time-dependent acoustic waves in a relaxing medium. A classical
solution of this problem is obtained in the form of convolutions
of the right-hand side and the initial data with the fundamental
solution of the equation. Sharp time estimates are deduced for
the solution in question which show polynomial growth for small
times and exponential decay for large time when f=0. They also
show the time evolution of the solution when f≠0.