Nonlocal filtration equations with rough kernels

We study the nonlinear and nonlocal Cauchy problem ∂tu+Lφ(u)=0inRN×R+,u(⋅,0)=u0, where LL is a Lévy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity φφ is nondecreasing and continuous, and the initial datum u0u0 is assumed to be in L1(RN)L1(RN). We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, φ(u)=∣u∣m−1uφ(u)=∣u∣m−1u, m>1m>1, these solutions turn out to be bounded and Hölder continuous for t>0t>0. We also describe the large time behaviour when the nonlinearity resembles a power for u≈0u≈0 and the kernel associated to LL is close at infinity to that of the fractional Laplacian.