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We derive general bounds for the large time size of supnorm values ofsolutions to one-dimensional advection-diffusion equations with initial data inLp0 (R) \cap L1 for some 1 <= p0 < \infty, and arbitrary bounded advectionspeeds b(x, t), introducing techniques based on suitable energy arguments. Someopen problems and related results are also given.

General asymptotic supnorm estimates for solutions of one-dimensional advection-diffusion equations in heterogeneous media, I