We derive general bounds for the large time size of supnorm values ‖𝑢(⋅, 𝑡)‖𝐿∞(R) of solutions to one-dimensional advectiondiffusion equations 𝑢𝑡 + (𝑏(𝑥, 𝑡)𝑢)𝑥 = 𝑢𝑥𝑥, 𝑥 ∈ R, 𝑡 > 0 with initial data 𝑢(⋅, 0) ∈ 𝐿𝑝 0 (R) ∩ 𝐿∞(R) for some 1 ≤ 𝑝0 < ∞and arbitrary bounded advection speeds 𝑏(𝑥, 𝑡), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.
International journal of partial differential equations. New York. Vol. 2014 (2014), 8 f. Article ID 450417