Nonlocal filtration equations with rough kernels

We study the nonlinear and nonlocal Cauchy problem partial derivative(t)u + L phi(u) = 0 in R-N x R+, u(center dot, 0) = u(0). where L is a Levy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity phi is nondecreasing and continuous, and the initial datum u(0) is assumed to be in L-1(R-N). We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, phi(u) =|u| m(-1) u, m > 1, these solutions turn out to be bounded and Holder continuous for t > 0. We also describe the large time behaviour when the nonlinearity resembles a power for u approximate to 0 and the kernel associated to L is close at infinity to that of the fractional Laplacian. (c) 2016 Elsevier Ltd. All rights reserved.

nonlinear nonlocal diffusion; regularity; asymptotic behavior; porous medium equation; symmetric jump processes; parabolic differential equations; dirichlet forms; intermediate asymptotics; fractional diffusion; operators; boundary

Nonlocal filtration equations with rough kernels Articles