A nonlocal diffusion equation on manifolds

MARIA DEL MAR GONZALEZ; CATHERINE BANDLE; NOEMI WOLANSKI; MARCO FONTELOS

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS

TAYLOR & FRANCIS INC

Lugar: Londres; Año: 2018 vol. 43 p. 652 - 676

0360-5302

In this paper we study a nonlocal diffusion problem on a manifold. These kinds of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.