International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 17, Pages 1073-1081

Time estimates for the Cauchy problem for a third-order hyperbolic equation

Vladimir Varlamov

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.


A classical solution is considered for the Cauchy problem: (uttΔu)t+uttαΔu=f(x,t), x3, t>0; u(x,0)=f0(x), ut(x,0)=f1(x), and utt(x)=f2(x), x3, 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 f0.