  We study integrability conditions for existence and nonexistence of a
local-in-time integral solution of fractional semilinear heat equations with
rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main
results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We
introduce a new supersolution which plays a crucial role. Our method does not
rely on a change of variables, and hence it can be applied to a wide class of
nonlocal parabolic equations. In particular, when the nonlinear term is $u^p$
or $e^u$, a local-in-time solution can be constructed in the critical case, and
integrability conditions for the existence and nonexistence are completely
classified. Our analysis is based on the comparison principle, Jensen's
inequality and $L^p$-$L^q$ type estimates.
