Weighted [formula omitted] estimates and Fujita exponent for a nonlocal equation

Abstract

We investigate a nonlocal equation ∂tu=∫RnJ(x−y)u(y,t)dy−‖J‖L1u(x,t)+a(x,t)up in Rn, where a is unbounded and J belongs to a weighted space. Crucial weighted Lp and interpolation estimates for the Green operator are established by a new method based on the sharp Young’s inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions Γ=(1+|x|2∕η)b∕2 that −Γ∕η≲J∗Γ−Γ≲Γ∕η and η−b+∕2≲Γ∕xb≲η−b−∕2. Blow-up behaviors are investigated by employing test functions ϕR=Γ (η=R) instead of principal eigenfunctions. Global well-posedness in weighted Lp spaces for the Cauchy problem is proved. When a∼xσ the Fujita exponent is shown to be 1+(σ+2)∕n. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.