Weighted estimates and Fujita exponent for a nonlocal equation

Abstract

We investigate a nonlocal equation ${\partial }_{t}u={\int }_{{\mathbb{R}}^n}J(x-y)(u(y,t)-u(x,t))dy+a(x,t)u^p$ 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 $\Gamma =(1+|x{|}^2/\eta {)}^{b/2}$ that $-\Gamma /\eta \lesssim J\ast \Gamma -\Gamma \lesssim \Gamma /\eta$ and ${\eta }^{-b_{+}/2}\lesssim \Gamma /{⟨x⟩}^b\lesssim {\eta }^{-b_{-}/2}$. Blow-up behaviors are investigated by employing test functions ${\varphi }_R=\Gamma$ ($\eta =R$) 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 $1+(\sigma +2)/n$. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.

arXiv comment30 pages

References in corpus (1)

• On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators