Doubly exponential growth and decay for a semilinear heat equation with logarithmic nonlinearity

In this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. (J. Math. Anal. Appl. 2015, 422, 84-98) and Han (J. Math. Anal. Appl. 2019, 474, 513-517). On the one hand, we not only prove the existence of doubly exponential decay solutions, but also find its threshold, and obtain the solutions with ∥ u 0 ∥ 2 2 → 0 + is always zero. On the other hand, we also prove the existence of doubly exponential growth solutions. The reseach results in this note extend previous results from both decay and growth.