We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely,The problem is posed in {x ∈ R N , t ∈ R} with nonnegative initial data u(x, 0) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. Here we establish the boundedness and C α regularity of such weak solutions
In this work we consider the problemswhere L is a nonlocal differential operator and Ω is a bounded domain in R N , with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summability of the solution u with respect to the summability of the datum f . In the process we establish an L p -theory, for p 1, associated to these problems and we prove some useful inequalities for the applications.2010 Mathematics Subject Classification. 45K05, 47G20, 35R09, 35D30, 35D35.
In this work we study the following fractional critical problemwhere Ω ⊂ R n is a regular bounded domain, λ > 0, 0 < s < 1 and n > 2s. Here (−∆) s denotes the fractional Laplace operator defined, up to a normalization factor, byOur main results show the existence and multiplicity of solutions to problem (P λ ) for different values of λ. The dependency on this parameter changes according to whether we consider the concave power case (0 < q < 1) or the convex power case (1 < q < 2 * s −1). These two cases will be treated separately.
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