Weighted estimates and Fujita exponent for a nonlocal equation

TL;DR

This paper studies a nonlocal PDE with unbounded coefficients, establishing weighted estimates, blow-up behavior, and the Fujita exponent, thereby unifying nonlocal diffusion and pseudoparabolic equations.

Contribution

It introduces a new method for weighted estimates using sharp Young's inequality and auxiliary functions, and determines the Fujita exponent for equations with unbounded coefficients.

Findings

Established weighted $L^p$ estimates for the Green operator.

Proved global well-posedness in weighted spaces.

Determined the Fujita exponent as $1+(\sigma+2)/n$ for certain coefficient growth.

Abstract

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.