Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem

C. Esteve, J.D. Rossi, A. San Antolín. Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem. BOUND VALUE PROBL (2014), pp. 109. DOI: 10.1186/1687-2770-2014-109

Abstract. We obtain upper bounds for the decay rate for solutions to the nonlocal problem $\partial_{t} u (x, t) = \int_{R^{n}} J (x, y) {| u (y, t) - u (x, t) |}^{p - 2} (u (y, t) - u (x, t)) d y$ with an initial condition $u_{0} \in L^{1} (R^{n}) \cap L^{\infty} (R^{n})$ and a fixed $p > 2$ . We assume that the kernel J is symmetric, bounded (and therefore there is no regularizing effect) but with polynomial tails, that is, we assume a lower bounds of the form $J (x, y) \geq c_{1} {| x - y |}^{- (n + 2 σ)}$ , for $| x - y | > c_{2}$ and $J (x, y) \geq c_{1}$ , for $| x - y | \leq c_{2}$ . We prove that ${∥ u (\cdot, t) ∥}_{L^{q} (R^{n})} \leq C t^{- \frac{n}{(p - 2) n + 2 σ} (1 - \frac{1}{q})}$ for $q \geq 1$ and t large.

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Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem

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