Abstract: This work is concerned with the asymptotic stability of traveling waves for scalar conservation laws with a convex flux function and a dispersion term. First we prove the existence of solutions locally in time of the initial value problem for initial data near a constant solution by Fourier analysis. using the semigroup method the local existence for initial data that are an L2-perturbation of a traveling wave profile is proved. We also obtain a regularity property of these solutions. The solution operator generated by the linearized equation plays a crucial role. Using the energy method we establish a priori estimates. These estimates, when combined with the local existency, lead to the desired global in time existence as well as the time-asymptotic decay of solutions with inital data close to a monotone traveling wave.
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