Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self‐similar solution...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2020-05, Vol.43 (8), p.5532-5563
Main Author: Fukuda, Ikki
Format: Article
Language:English
Subjects:
asymptotic profile
Asymptotic properties
Boundary value problems
Burgers equation
Decay rate
Diffusion rate
Diffusion waves
hyperbolic relaxation system
nonlinear damped wave equation
optimal decay estimate
second asymptotic profile
slowly decaying data
Wave equations
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Summary:We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self‐similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.6295