Chasseigne, E.; Felmer, P.; Rossi, J. D.; Topp, E. Fractional decay bounds for nonlocal zero order heat equations. (English) Zbl 1302.35052 Bull. Lond. Math. Soc. 46, No. 5, 943-952 (2014). Summary: In this paper, we obtain bounds for the decay rate for solutions to the nonlocal problem \({\partial}_t u(t,x) = \int _{\mathbb {R} ^n} J(x,y)[u(t,y) - u(t,x)]\,dy\). Here we deal with bounded kernels \(J\) but with polynomial tails, that is, we assume a lower bound of the form \(J(x,y) \geq c_1|x-y|^{-(n +2\sigma )}\), for \(|x - y| >c_2\). Our estimates takes the form \(\|u(t)\|_{L^q(\mathbb {R} ^n)} \leq C t^{-({n}/{2\sigma }) (1 - {1}/{q})}\) for \(t\) large. Cited in 7 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 45P05 Integral operators 35R09 Integro-partial differential equations Keywords:polynomial tails PDFBibTeX XMLCite \textit{E. Chasseigne} et al., Bull. Lond. Math. Soc. 46, No. 5, 943--952 (2014; Zbl 1302.35052) Full Text: DOI arXiv References: [1] Andreu-Vaillo, Nonlocal diffusion problems (2010) · Zbl 1214.45002 · doi:10.1090/surv/165 [2] C. Brändle A. de Pablo Nonlocal heat equations: decay estimates and Nash inequalities [3] Chasseigne, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. 86 ((9)) pp 271– (2006) · Zbl 1126.35081 · doi:10.1016/j.matpur.2006.04.005 [4] Di Nezza, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 pp 521– (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004 [5] Ignat, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl. 92 ((9)) pp 163– (2009) · Zbl 1173.35363 · doi:10.1016/j.matpur.2009.04.009 [6] Stroock, An introduction to the theory of large deviations (1984) · Zbl 0552.60022 · doi:10.1007/978-1-4613-8514-1 [7] Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 pp 240– (1985) · Zbl 0608.47047 · doi:10.1016/0022-1236(85)90087-4 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.