Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

Abstract

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu _t +f(u) _x =μu _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₊)<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 ast→∞. 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.