Stability of Traveling Waves of a Solute Transport Equation

We prove the large time asymptotic stability of traveling wave solutions to the scalar solute transport equation (contaminant transport equation) with spatially periodic diffusion-adsorption coefficients in one space dimension. The time dependent solutions converge in proper norms to a translate of traveling wave solutions as time approaches infinity. In case of classical traveling waves, the convergence rate is exponential in time for a class of small initial perturbations; and for general order one perturbations, the convergence holds in supremum norm. In case of degenerate Hölder continuous traveling waves, the convergence holds inL1norm. As a byproduct, uniqueness up to translation of degenerate traveling waves follows. We use maximum principle,L1contraction, spectral theory, and a space-time invariance property of solutions.