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Gamma-ray bursts (GRBs) are brief, intense pulses of high-energy emission associated with extreme astrophysical phenomena, e.g. the death of massive stars or the coalescence of compact objects. They have been observed at high energies by the Fermi Large Area Telescope (LAT), which detects GRBs in the 20 MeV–300 GeV energy range. The Fermi-LAT Second GRB Catalog (2FLGC) presents information on 186 GRBs observed from 2008 to 2018. We consider the GRBs that have been fitted in the 2FLGC with a broken (21 GRBs) or simple power law (65 GRBs), compiling a total sample of 86 GRBs. We analyze the relationship between the spectral and temporal indices using closure relations according to the synchrotron forward-shock model evolving in stratified environments (n∝r−k). We find that the model without energy injection is preferred over the one with energy injection. There is a clear preference for the cooling conditions ν> max{νc,νm} and νm<ν<νc (where νc and νm are the cooling and characteristic frequencies, namely the frequency at the spectral break). Within these cooling conditions, density profiles r−k with values of k=1.5 and 2 generally have a higher rate of occurrence when considering relations with and without energy injection.

X-ray flares are frequently detected in the X-ray afterglow light curves and are highly correlated with the prompt emission of gamma-ray bursts (GRBs). We compile a comprehensive sample of X-ray flares up to 2021 April, comprising 697 flares. We classify the total sample into four types: early flares (tp≤103 s), late flares (tp>103 s), long gamma-ray burst (LGRB) flares and short gamma-ray burst (SGRB) flares, and analyze the distributions and relationships of the flare parameters. It is found that the early flares have a higher frequency, shorter duration, and more asymmetrical structure. In addition, the distributions of the morphological parameters of the SGRB flares are similar to those of the LGRB flares. We also find that the durations and rising (decay) times of the early flares are positively correlated with the peak times, but the late flares follow the different dependent relations. There is a strong anti-correlation between the peak luminosities (LX,P) and the peak times of the flares, e.g., LX,p∝tp,z−1.45 for the LGRB flares, and LX,p∝tp,z−1.27 for the SGRB flares, respectively. Furthermore, the peak luminosity is highly dependent on the isotropic energy (EX,iso) for the early LGRB flares, the best fit is LX,p∝EX,iso1.06(r=0.89). We also find a tight three-parameter correlation, LX,p∝tp,z−1.03EX,iso0.92(r=0.96). All the late flares fall into the 3σ confidence region defined by the early flares. In terms of the point of kinematic arguments, both the SGRB and LGRB flares support a common scheme of internal origin. The SGRB flares have similar properties to the LGRB flares, suggesting that both of them share a similar physical mechanism from the late-time activity of central engine.