We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the longtime behavior is identified by means of a nonlocal mesa solving an obstacle problem.

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: ut = J * u − u , where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.

No abstract

No abstract

The paper deals with the asymptotic behaviour of solutions to the porous media equation, u t = ∆u m , m > 1, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on ∂Ω. When the space dimension is three or more this behaviour is given by a Barenblatt function away from the fixed boundary ∂Ω and near the free boundary. The asymptotic behaviour of the free boundary is given by the same Barenblatt function. On the other hand, if the solution is scaled according to its decay factor, away from the free boundary and close to the holes it behaves like a function whose m-th power is harmonic and vanishes on ∂Ω. The height of such a function is determined by matching with the Barenblatt solution representing the outer behaviour. The inner and the outer behaviour can be presented in a unified way through a suitable global approximation.

We consider the Cauchy Problem for the class of nonlinear parabolic equations of the form u t = a(u)∆u + |∇u| 2 , with a function a(u) that vanishes at u = 0. Because of the degenerate character of the coefficient a the usual concept of viscosity solution in the sense of Crandall-Evans-Lions has to be modified to include the behaviour at the free boundary. We prove that the problem is well-posed in a suitable class of viscosity solutions. Agreement with the concept of weak solution is also shown.

We study the short and large time behaviour of solutions of nonlocal heat equations of the form ∂tu + Lu = 0. Here L is an integral operator given by a symmetric nonnegative kernel of Lévy type, that includes bounded and unbounded transition probability densities. We characterize when a regularizing effect occurs for small times and obtain L q -L p decay estimates, 1 ≤ q < p < ∞ when the time is large. These properties turn out to depend only on the behaviour of the kernel at the origin or at infinity, respectively, without need of any information at the other end. An equivalence between the decay and a restricted Nash inequality is shown. Finally we deal with the decay of nonlinear nonlocal equations of porous medium type ∂tu + LΦ(u) = 0.2000 Mathematics Subject Classification. Primary: 45P05, 35S10, 45A05; Secondary: 45M05, 45G10.