General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

摘要

We derive general bounds for the large time size of supnorm values∥u(·,t)∥L∞(ℝ)of solutions to one-dimensional advection-diffusion equationsut+(b(x,t)u)x=uxx,x∈ℝ,t>0with initial datau(·,0)∈Lp0(ℝ)∩L∞(ℝ)for some1≤p0<∞and arbitrary bounded advection speedsb(x,t), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.