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

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)|p−2(u(y,t)−u(x,t))dy with an initial condition u0∈L1(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|x−y|−(n+2σ), for |x−y|>c2 and J(x,y)≥c1, for |x−y|≤c2. We prove that ∥u(⋅,t)∥Lq(Rn)≤Ct−n(p−2)n+2σ(1−1q) for q≥1 and t large.