2014/12/18 by F. Ali Mehmeti, Mehmeti, F. Ali, F. Dewez +1 · 2 citations
Mathematics · #35B30 #35B40 #35Q41 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 41A80 #Secondary 41A60 #math.AP #msc:35B30 #msc:35B40 #msc:35Q41 #msc:41A60 #msc:41A80
paper · pdf · doi:10.48550/arxiv.1412.5792
This paper is part II of a series of two articles. The title of the first part is: "Explicit error estimates for the stationary phase method I: The influence of amplitude singularities"
arxiv created 2014/12/18 · arxiv updated 2014/12/19
In this paper, we improve slightly Erdélyi's version of the stationary phase method by replacing the employed smooth cut-off function by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain uniform estimates of the solution in space-time regions which are asymptotically larger than any space-time cones. Moreover we expand the solution with respect to time on the boundaries of the above regions, showing the optimality of the decay rates of the estimates.