A Framework for Nonlocal, Nonlinear Initial Value Problems

Karch G, Kaßmann M, Krupski M. A Framework for Nonlocal, Nonlinear Initial Value Problems. SIAM Journal on Mathematical Analysis. 2020;52(3):2383-2410.We study the Cauchy problem for nonlinear, nonlocal operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g., the porous medium equation with the fractional Laplacian and the parabolic fractional p-Laplacian. We show the existence and uniqueness of bounded solutions, and study their further properties. Several new examples of nonlocal, nonlinear operators are provided