Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity |
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Authors: | Akitaka Matsumura Kenji Nishihara |
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Institution: | (1) Department of Mathematics, Osaka University, 560 Osaka, Japan;(2) School of Political Science and Economics, Waseda University, 169-50 Tokyo, Japan |
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Abstract: | The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu
t
+f(u)
x
= u
xx
with the initial datau
0 which tend to the constant statesu
± asx ± . Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f (u
±). Moreover, the rate of asymptotics in time is investigated. For the casef (u+)(u–), if the integral of the initial disturbance over (– ,x) is small and decays at the algebraic rate as |x|![rarr](/content/phg0752022537814/xxlarge8594.gif) , then the solution approaches the traveling wave at the corresponding rate ast![rarr](/content/phg0752022537814/xxlarge8594.gif) . This rate seems to be almost optimal compared with the rate in the casef=u
2/2 for which an explicit form of the solution exists. The rate is also obtained in the casef (u
±
=s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures. |
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Keywords: | |
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