  We investigate a nonlocal equation
$\partial_tu=\int_{\mathbb{R}^n}J(x-y)(u(y,t)-u(x,t))dy+a(x,t)u^p$ 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 $\varGamma=(1+|x|^2/\eta)^{b/2}$ that
$-\varGamma/\eta\lesssim J\ast\varGamma-\varGamma\lesssim\varGamma/\eta$ and
$\eta^{-b_+/2}\lesssim\varGamma/\left\langle x\right\rangle^b\lesssim
\eta^{-b_-/2}$. Blow-up behaviors are investigated by employing test functions
$\phi_R=\varGamma$ ($\eta=R$) instead of principal eigenfunctions. Global
well-posedness in weighted $L^p$ spaces for the Cauchy problem is proved. When
$a\sim\left\langle x\right\rangle^\sigma$ the Fujita exponent is shown to be
$1+(\sigma+2)/n$. Our approach generalizes and unifies nonlocal diffusion
equations and pseudoparabolic equations.
