Nonlocal filtration equations with rough kernels

We study the nonlinear and nonlocal Cauchy problem \[ \partial_{t}u+\mathcal{L}\varphi(u)=0 \quad\text{in }\mathbb{R}^{N}\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0, \] where $\mathcal{L}$ is a L\'evy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity $\varphi$ is nondecreasing and continuous, and the initial datum $u_0$ is assumed to be in $L^1(\mathbb{R}^N)$. We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, $\varphi(u)=|u|^{m-1}u$, $m>1$, these solutions turn out to be bounded and H\"older continuous for $t>0$. We also describe the large time behaviour when the nonlinearity resembles a power for $u\approx 0$ and the kernel associated to $\mathcal{L}$ is close at infinity to that of the fractional Laplacian.