We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion,We consider data f ∈ L 1 (R N ) and all exponents 0 < σ < 2 and m > 0. Existence and uniqueness of a weak solution is established for m > m * = (N − σ) + /N , giving rise to an L 1 -contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range 0 < m ≤ m * existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above m * . We also study the dependence of solutions on f, m and σ. Moreover, we consider the above questions for the problem posed in a bounded domain.2000 Mathematics Subject Classification. 26A33, 35A05, 35K55, 76S05

Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients through vasculature. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem.Our first goal here is to formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain limit. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the asymptotic Hele-Shaw type problem. The main tools are nonlinear regularizing effects for certain porous medium type equations, regularization techniquesà la Steklov, and a Hilbert duality method for uniqueness. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. A complete description requires the equation on the cell number density.Using this theory as a basis, we go on to consider the more complex model including nutrients. We obtain the equation for the limit of the coupled system; the method relies on some BV bounds and space/time a priori estimates. Here, new technical difficulties appear, and they reduce the generality of the results in terms of the initial data. Finally, we prove uniqueness for the system, a main mathematical difficulty.

We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion,with m > m * = (N − 1)/N , N ≥ 1 and f ∈ L 1 (R N ). An L 1 -contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all x ∈ R N , t > 0.

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusionIf the nonlinearity satisfies some not very restrictive conditions: ϕ ∈ C 1,γ (R), 1 + γ > σ, and ϕ ′ (u) > 0 for every u ∈ R, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the non-linearity to obtain higher regularity for the solutions, even C ∞ regularity. Degenerate and singular cases, including the power nonlinearity ϕ(u) = |u| m−1 u, m > 0, are also considered, and the existence of classical solutions in the power case is proved.

We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit.Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of 'mushy region' prepares the tumor invasion.