General asymptotic supnorm estimates for solutions of one-dimensional advection-diffusion equations in heterogeneous media
Summary: We derive general bounds for the large time size of supnorm values \(\| u(\cdot,t)\|_{L^\infty(\mathbb R)}\) of solutions to one-dimensional advection-diffusion equations \(u_t+(b(x,t)u)_x=u_{xx}\), \(x\in\mathbb R\), \(t>0\) with initial data \(u(\cdot,0)\in L^{p_0}(\mathbb R)\cap L^\infty(\mathbb R)\) for some \(1\leq p_0<\infty\) 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.
scientific article; zbMATH DE number 6349586