The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation laws ut+f(u)x=μuxx with the initial data u0 which tend to the constant states u± as x→±∞. Stability theorems are obtained in the absence of the convexity of f and in the allowance of s (shock speed)=f′(u±). Moreover, the rate of asymptotics in time is investigated. For the case f′(u+)<s<f′(u-), if the integral of the initial disturbance over (-∞, x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate as t→∞. This rate seems to be almost optimal compared with the rate in the case f=u2/2 for which an explicit form of the solution exists. The rate is also obtained in the case f′(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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics