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 supersolution of an integral equation which can be applied to a nonlocal parabolic equation. 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.

我们研究了在均匀局部\（L ^ p \）空间中具有相当普遍增长的非线性的分数阶半线性热方程的时间局部积分解的存在性和不存在性的可积性条件。我们对此问题的主要结果包括定理1.4、1.6、5.1和5.3。我们介绍了积分方程的超解，该积分方程可以应用于非局部抛物线方程。当非线性项为\（u ^ p \）或\（e ^ u \）时，可以在临界情况下构造局部时间解，并且将存在和不存在的可积性条件完全分类。我们的分析基于比较原理，詹森不等式和\（L ^ p \） - \（L ^ q \）类型估计。