Time estimates for the Cauchy problem for a third-order hyperbolic equation
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 o...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203204361 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |