A necessary and sufficient conditions for the global existence of solutions to fractional reaction-diffusion equations on RN.

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations u t = Δ α u + ψ (t) f (u) , in R N × (0 , ∞) , u (· , 0) = u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades, where N ≥ 2 , Δ α = - - Δ α / 2 denotes the fractional Laplace operator with 0 < α ≤ 2 , ψ is a nonnegative and continuous function, and f is a convex function. The purpose of this paper is to resolve this problem completely as follows: There is a global solution to the equation if and only if ∫ 1 ∞ ψ (t) t N α f ϵ t - N α d t < ∞ , for some ϵ > 0. [ABSTRACT FROM AUTHOR]