Nonlocal filtration equations with rough kernels. (English) Zbl 1334.35409
Summary: We study the nonlinear and nonlocal Cauchy problem
\[
\partial_t u + \mathcal{L} \varphi(u) = 0 \text{in} \mathbb{R}^N \times \mathbb{R}_+, u(\cdot, 0) = u_0,
\]
where \(\mathcal{L}\) is a Lévy-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, \(\varphi(u) = \mid u \mid^{m-1} u\), \(m > 1\), 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 \(u \approx 0\) and the kernel associated to \(\mathcal{L}\) is close at infinity to that of the fractional Laplacian.
MSC:
| 35R11 | Fractional partial differential equations |
| 35S10 | Initial value problems for PDEs with pseudodifferential operators |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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