Approximation of random diffusion equations by nonlocal diffusion equations in free boundary problems of one space dimension

Abstract

<p>We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, <i>J. Functional Anal</i>. <b>277</b> (2019) 2772–2814" C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, <i>Interfaces Free Bound</i>. <b>21</b> (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form Jϵ(x)=1ϵJ(xϵ) for small ϵ>0 , where J(x) has compact support. We also give an estimate of the error term of the approximation by some positive power of ϵ.</p>