Abstract
We study the nonlinear and nonlocal Cauchy problem ∂tu+Lφ(u)=0inRN×R+,u(⋅,0)=u0, where L is a Lévy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity φ is nondecreasing and continuous, and the initial datum u0 is assumed to be in 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−1u, m>1, these solutions turn out to be bounded and Hölder continuous for t>0. We also describe the large time behaviour when the nonlinearity resembles a power for u≈0 and the kernel associated to L is close at infinity to that of the fractional Laplacian.
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