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

Let \(0<\theta \le 2\) $0<\theta \le 2$ , \(N\ge 1\) $N\ge 1$ and \(T>0\) $T>0$ . We are concerned with the Cauchy problem for the fractional semilinear parabolic equation \(\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+(-\Delta )^{\theta /2}u=f(u) & \text {in}\ {{\mathbb {R}}^N}\times (0,T),\\ u(x,0)=u_0 (x)\ge 0 & \text {in}\ {{\mathbb {R}}^N}. \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u+{(-\mathrm{\Delta })}^{\theta /2}u=f(u)\hfill & \text{in}{\mathbb{R}}^N\times (0,T),\hfill \\ u(x,0)=u_0(x)\ge 0\hfill & \text{in}{\mathbb{R}}^N.\hfill \end{array}\right)\end{array}$$ Here, \(f\in C[0,\infty )\) $f\in C[0,\infty )$ denotes a rather general growing nonlinearity and \(u_0\) $u_0$ may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when \(f(u)=u^{1+{\theta }/{N}}\left[ \log (u+e)\right] ^{a}\) $f(u)=u^{1+\theta /N}{\left[log,(,u,+,e,)\right]}^a$ , we obtain a sharp integrability condition on \(u_0\) $u_0$ which explicitly determines local in time existence/nonexistence of a nonnegative solution.