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 tu(x,t)=RnJ(x,y)|u(y,t)u(x,t)|p2(u(y,t)u(x,t))dy with an initial condition u0L1(Rn)L(Rn) 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)c1|xy|(n+2σ), for |xy|>c2 and J(x,y)c1, for |xy|c2. We prove that u(,t)Lq(Rn)Ctn(p2)n+2σ(11q) for q1 and t large.