Abstract

We study the general family of nonlinear evolution equations of fractional diffusive type ∂tu− div (|u|m1∇(−Δ)−s[|u|m2−1u])=f. Such nonlocal equations are related to the porous medium equations with a fractional Laplacian pressure. Our study concerns the case in which the flow takes place in the whole space. We consider m1,m2>0, and s∈(0,1), and prove the existence of weak solutions. Moreover, when f≡0 we obtain the Lp-L∞ decay estimates of solutions, for p≧1. In addition, we also investigate the finite time extinction of solution.