Energy and large time estimates for nonlinear porous medium ow with nonlocal pressure in RN

Abstract

We study the general family of nonlinear evolution equations of fractional diffusive type [delta]u-div(|u|m1[nabla]([delta]-s||u||m2-1u|= f. Such type of nonlocal equationsare related to the porous medium equations with a fractional Laplacian pressure.Our study concerns the case in which the ow takes place in the whole space. We consider m1;m2 > 0, and s 2 (0; 1), and prove existence of weak solutions. Moreover, when f _ 0 we obtain the Lp-L1 decay estimates of solutions, for p _ 1. Besides, we also investigate the _nite time extinction of solution. Our results improve the recent papers in the literature.