Solvability of the Cauchy problem for fractional semilinear parabolic equations in critical and doubly critical cases

Let \(0<\theta \le 2\), \(N\ge 1\) and \(T>0\). We are concerned with the Cauchy problem for the fractional semilinear parabolic equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+(-\Delta )^{\theta /2}u=f(u) &{} \text {in}\ {{\mathbb {R}}^N}\times (0,T),\\ u(x,0)=u_0 (x)\ge 0 &{} \text {in}\ {{\mathbb {R}}^N}. \end{array}\right. } \end{aligned}$$

Here, \(f\in C[0,\infty )\) denotes a rather general growing nonlinearity and \(u_0\) may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when \(f(u)=u^{1+{\theta }/{N}}\left[ \log (u+e)\right] ^{a}\), we obtain a sharp integrability condition on \(u_0\) which explicitly determines local in time existence/nonexistence of a nonnegative solution.