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

In this paper, we consider the fractional heat equation u_t=△^α/2u+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 u₀∈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→∞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)=sup_1≤t≤sf(t)/t. When lim_s→0f(s)/s<∞, the same characterisations holds for the fractional heat equation on the whole space R^d.