A characteristic of local existence for nonlinear fractional heat equations in Lebesgue spaces

In this paper, we consider the fractional heat equation ut=△α/2u+f(u) with Dirichlet conditions on the ball BR⊂Rd, 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 Lq(BR) provided that the non-negative initial data u0∈Lq(BR). For q>1 and 1<α<2, we show that the equation has a local solution in Lq(BR) if and only if lims→∞sups−(1+αq/d)f(s)=∞; and for q=1 and 1<α<2 if and only if ∫1∞s−(1+α/d)F(s)ds<∞, where F(s)=sup1≤t≤sf(t)/t. When lims→0f(s)/s<∞, the same characterisations holds for the fractional heat equation on the whole space Rd.