Decay estimates for nonlocal problems via energy methods

In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form u t ( x , t ) = ∫ R d G ( x − y ) ( u ( y , t ) − u ( x , t ) ) d y . For example, we will consider equations like, u t ( x , t ) = ∫ R d J ( x , y ) ( u ( y , t ) − u ( x , t ) ) d y + f ( u ) ( x , t ) , and a nonlocal analogous to the p-Laplacian, u t ( x , t ) = ∫ R d J ( x , y ) | u ( y , t ) − u ( x , t ) | p − 2 ( u ( y , t ) − u ( x , t ) ) d y . The energy method developed here allows us to obtain decay rates of the form, ‖ u ( ⋅ , t ) ‖ L q ( R d ) ⩽ C t − α , for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration.