Weighted Lp estimates and Fujita exponent for a nonlocal equation. (July 2019) - Khomrutai, Sujin
Abstract: We investigate a nonlocal equation ∂ t u = ∫ R n J ( x − y ) u ( y, t ) d y − ‖ J ‖ L 1 u ( x, t ) + a ( x, t ) u p in 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 Γ = ( 1 + | x | 2 ∕ η ) b ∕ 2 that − Γ ∕ η ≲ J ∗ Γ − Γ ≲ Γ ∕ η and η − b + ∕ 2 ≲ Γ ∕ x b ≲ η − b − ∕ 2 . Blow-up behaviors are investigated by employing test functions ϕ R = Γ ( η = R ) instead of principal eigenfunctions. Global well-posedness in weighted L p spaces for the Cauchy problem is proved. When a ∼ x σ the Fujita exponent is shown to be 1 + ( σ + 2 ) ∕ n . Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.