ASYMPTOTIC STABILITY OF TRAVELING WAVES FOR SCALAR VISCOUS CONSERVATION-LAWS WITH NONCONVEX NONLINEARITY

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation laws u(t) + f(u)x = muu(xx) with the initial data u0 which tend to the constant states u+/- as x --> +/- infinity. 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 (- infinity, x) is small and decays at the algebraic rate as \x\ --> infinity, then the solution approaches the traveling wave at the corresponding rate as t --> infinity. 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.