Decay estimates for nonlinear nonlocal diffusion problems in the whole space

Abstract

In this paper, we obtain bounds for the decay rate in the Lr (ℝd)-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, ut(x,t)=∫RdK(x,y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy,x∈Rd,t>0. We consider a kernel of the form K(x, y) = ψ(y−a(x)) + ψ(x−a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x)=Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)|u(y)−u(x)|p−2(u(y)−u(x))dy,1⩽p<∞. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ℝd: λ1,p(Rd)=2(∫Rdψ(z)dz)|1|detA|1/p−1|p. Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1,p 1/p as p→∞.