Decay estimates of nonlocal diffusion equations in some particle systems

Abstract

In this paper, we first consider a nonlocal mixed diffusion equation ∂tu(t,x)=Lu(t,x) in multiparticle systems with different spatial distributions, where the combination of nonlocal diffusion operators L in a spatial variable is defined by the improper integrals with kernels Jj(x) and the low frequencies of Ĵj(ξ) have the same kind of asymptotic expansions. We use the energy method to establish the decay estimates of solutions to ∂tu(t,x)=Lu(t,x) with initial condition u(0, x) = u0(x). Finally, we consider an anisotropic single-particle equation with nonlocal operator L is defined by the improper integrals with kernels Jj(xj) and it is a generation of an anisotropic nonlocal operator −∑j=1N(−∂xjxj)αj.