Premium
Stability of travelling wave solutions to a semilinear hyperbolic system with relaxation
Author(s) -
Ueda Yoshihiro
Publication year - 2008
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.1044
Subject(s) - mathematics , exponential stability , traveling wave , relaxation (psychology) , stability (learning theory) , convergence (economics) , mathematical proof , mathematical analysis , disturbance (geology) , rate of convergence , shock wave , space (punctuation) , exponential growth , energy method , energy (signal processing) , initial value problem , geometry , physics , nonlinear system , mechanics , statistics , channel (broadcasting) , philosophy , economic growth , linguistics , computer science , engineering , biology , psychology , social psychology , paleontology , electrical engineering , economics , quantum mechanics , machine learning
We study a semilinear hyperbolic system with relaxation and investigate the asymptotic stability of travelling wave solutions with shock profile. It is shown that the travelling wave solution is asymptotically stable, provided the initial disturbance is suitably small. Moreover, we show that the time convergence rate is polynomially (resp. exponentially) fast as t →∞ if the initial disturbance decays polynomially (resp. exponentially) for x →∞. Our proofs are based on the space–time weighted energy method. Copyright © 2008 John Wiley & Sons, Ltd.