A characteristic of local existence for nonlinear fractional heat equations in Lebesgue spaces. (15th February 2017) - Li, Kexue
Abstract: In this paper, we consider the fractional heat equation u t = △ α / 2 u + f ( u ) with Dirichlet conditions on the ball B R ⊂ R d, where △ α / 2 is the fractional Laplacian, f : [ 0, ∞ ) → [ 0, ∞ ) is continuous and non-decreasing. We present the characterisations of f to ensure the equation has a local solution in L q ( B R ) provided that the non-negative initial data u 0 ∈ L q ( B R ) . For q > 1 and 1 < α < 2, we show that the equation has a local solution in L q ( B R ) if and only if lim s → ∞ sup s − ( 1 + α q / d ) f ( s ) = ∞ ; and for q = 1 and 1 < α < 2 if and only if ∫ 1 ∞ s − ( 1 + α / d ) F ( s ) d s < ∞, where F ( s ) = sup 1 ≤ t ≤ s f ( t ) / t . When lim s → 0 f ( s ) / s < ∞, the same characterisations holds for the fractional heat equation on the whole space R d .