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Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term
Author(s) -
Kato Masakazu,
Ueda Yoshihiro
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4561
Subject(s) - mathematics , term (time) , nonlinear system , mathematical analysis , burgers' equation , damped wave , convection , representation (politics) , wave equation , initial value problem , space (punctuation) , partial differential equation , physics , mechanics , quantum mechanics , linguistics , philosophy , politics , political science , law
This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one‐dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.