Local approximation of heterogeneous porous medium equation by some nonlocal dispersal problems

Abstract

The classical porous medium equation is widely used to model different natural phenomena related to diffusion, filtration and heat propagation. In this short communication, we prove that the solution of porous medium equation can be locally approximated by the solution of a class of nonlocal dispersal equation. Our work is a counterpart to the important works (see Berestycki et al. [J. Funct. Anal. 271 (2016), pp. 2701–2751; J. Math. Biol. 72 (2016), pp. 1693–1745]; Dipierro et al. [J. Eur. Math. Soc. (JEMS) 19 (2017), pp. 957–966; J. Geom. Anal. 29 (2019), pp. 1428–1455]; Hansen and Netuka [Potential Anal. 2 (1993), pp. 67–71]; Ignat and Rossi [J. Funct. Anal. 251 (2007), pp. 399–437]; Shen and Xie [J. Differential Equations 259 (2015), pp. 7375–7405]; Sprekels and Valdinoci [SIAM J. Control Optim. 55 (2017), pp. 70–93]).