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Abstract. In this paper we study the blow-up phenomenon for nonnegative solutions to the following parabolic problem:After discussing existence and uniqueness we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p + > 1.When Ω = R N we show that if p− > 1 + 2/N then there are global nontrivial solutions while if 1 < p − ≤ p + ≤ 1 + 2/N then all solutions to the problem blow up in finite time. Moreover, in case p− < 1+2/N < p+ there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global nontrivial solutions.When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough then the problem possesses global nontrivial solutions regardless the size of p(x).

In this paper we study numerical approximations of the nonlocal p−Laplacian type diffusion equation, u t (t, x) = Ω J(x − y)|u(t, y) − u(t, x)| p−2 (u(t, y) − u(t, x)) dy. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties of the continuos problem: it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity. Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties. In addition, we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile. Finally, we present some numerical experiments that illustrate our results.

Abstract. In this paper we study solutions to a nonlocal 1−laplacian equation given bywith u(x) = ψ(x) for x ∈ Ω J \ Ω. We introduce two notions of solutions and prove that the weakest of these two concepts of solution is equivalent to verify an equation involving the median of the function, that is, the value of u at a point x is the median of u in a ball centered at x (with a measure related J). We also show that solutions in the strongest sense are the nonlocal analogous to local least gradient functions, in the sense that they minimize a nonlocal functional. In addition, we prove that they converge to least gradient functions when the kernel J is appropriately rescaled.

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