Jean Velin : On a nonlinear nonlocal parabolic equation with Neumann boundary conditions
This paper deals with the existence of a unique classical solution for a parabolic quasi-linear problem involving the fractional p-Laplacian with Neumann boundary conditions, where the nonlinear source term depends on the solution. The approach used is based on Rothe's discretisation method under non classical assumptions. Besides, we demonstrate that this solution converges to a weak solution of the associated elliptic problem at a finite time. The existence of this solution in an appropriate fractional Sobolev space is also proven. Finally, we establish some additional properties, such as mass conservation and the extremum principle, for the solution of the parabolic problem. We also prove the boundedness of the solution of the associated elliptic problem using a De Giorgi iteration argument.