Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 14, pp. 1-62.

Title: Properties of the Dirichlet Green's function for linear diffusions on a half line

Authors: Joseph G. Conlon  (Univ. of Michigan, Ann Arbor, MI, USA)
         Michael Dabkowski (Univ. of Michigan, Dearborn, MI, USA)
  
Abstract:
This article  concerns the study of Green's functions for one dimensional 
diffusions with constant diffusion coefficient and  linear time inhomogeneous drift.
It is well know that the whole line Green's function is given by a Gaussian.
Formulas for the Dirichlet Green's function on the half line are only known 
in special cases.  The main object of study in the paper is the ratio 
of the Dirichlet to whole line Green's functions.  Bounds, asymptotic 
behavior in the limit as the diffusion coefficient vanishes, and 
a log concavity result are obtained for this ratio. These results have 
been used in the proof of asymptotic behavior for a simple model of 
Ostwald ripening.  


Submitted December 13, 2023. Published February 19, 2025.
Math Subject Classifications: 35F21, 35K20, 49N10.
Key Words: Nonlinear PDE; coarsening.