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RESEARCH PRODUCT

Fractional p-Laplacian evolution equations

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Abstract In this paper we study the fractional p-Laplacian evolution equation given by u t ( t , x ) = ∫ A 1 | x − y | N + s p | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y for  x ∈ Ω ,  t > 0 , 0 s 1 , p ≥ 1 . In a bounded domain Ω we deal with the Dirichlet problem by taking A = R N and u = 0 in R N ∖ Ω , and the Neumann problem by taking A = Ω . We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u ( y ) − u ( x ) | u ( y ) − u ( x ) | when u ( y ) = u ( x ) . We also consider the Cauchy problem in the whole R N by taking A = Ω = R N . We find existence and uniqueness of strong solutions for each of the above mentioned problems. We also study the asymptotic behaviour of these solutions as t → ∞ . Finally, we recover the local p-Laplacian evolution equation with Dirichlet or Neumann boundary conditions, and for the Cauchy problem, by taking the limit as s → 1 in the nonlocal problems multiplied by a suitable scaling constant.