A Widder's Type Theorem for the Heat Equation with Nonlocal Diffusion

摘要

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem ut + (-△)α/2u =0 for(x, t) ∈ Rn × (0, T), 0 < α < 2, can be written as u(x, t) =∫_R~n P_t (x - y)u(y, 0) dy, where P_t (x) =1/t~(n/α)P(χ/t~(1/α)), and P(x) :=∫_R~n e~(i x·ξ?|ξ |α) dξ. This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.