Weighted estimates and Fujita exponent for a nonlocal equation

We investigate a nonlocal equation ${\mathrm{\partial }}_{t}u={\int }_{{\mathbb{R}}^n}J(x-y)(u(y,t)-u(x,t))dy+a(x,t)u^p$ in ${\mathbb{R}}^n$, where $a$ is unbounded and $J$ belongs to a weighted space. Crucial weighted $L^p$ 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 $\Gamma =(1+\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}\eta {)}^{b\mathrm{/}2}$ that $-\Gamma \mathrm{/}\eta \lesssim J\ast \Gamma -\Gamma \lesssim \Gamma \mathrm{/}\eta$ and $\eta^{-b_+/2}\lesssim\varGamma/\left\langle x\right\rangle^b\lesssim \eta^{-b_-/2}$\mathrm{.}Blow-upbehaviorsareinvestigatedbyemployingtestfunctions$\phi_R=\varGamma$($\eta=R$) instead of principal eigenfunctions. Global well-posedness in weighted $L^p$ spaces for the Cauchy problem is proved. When $a\sim {⟨x⟩}^{\sigma }$ the Fujita exponent is shown to be $1+(\sigma +2)\mathrm{/}n$. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.