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

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.