Nonlocal equations with regular varying decay solutions

Abstract

We present a new approach for studying the large time asymptotic behavior to the nonlocal equation ∂tu=Ju−χ0u in the whole Rn, where Ju=∫RnJ(⋅,y)u(y,t)dy. We obtain a condition so that the solutions decay in time at the rate of a regular varying function R, even for non-symmetric J, for a wide class of R. This condition involves a sharp bound for Jku0 (u0=u|t=0) or the kernels of Jk, and the decay rate is proved by analyzing an exponential series with regular varying coefficients. Classical cases, including the fractional Laplacian, the compactly support regular kernels, and algebraic decaying kernels are shown to satisfy our condition by employing the sharp Young's inequality and the Fourier transform technique. In particular, we have generalized the pioneer work by Chasseigne et al. (2006) [11]. Finally, we exhibit nonlocal equation that has solutions with a prescribed decay rate, for a wide class of regular varying function.