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.
Categories: math.AP