Boudjeriou, TahirVan Thin, Nguyen2024-10-282024-10-2820241468-1218https://doi.org/10.1016/j.nonrwa.2024.10409https://www.sciencedirect.com/science/article/abs/pii/S1468121824000397https://dspace.univ-boumerdes.dz/handle/123456789/14557In this paper, we discuss some qualitative analysis of solutions to the following Cauchy problem of wave equations involving the 1/2-Laplace operator with critical exponential nonlinearity [Formula presented] where λ>0, δ≥0, q>2, and α0>0. By using the contraction mapping principle, we show that the above Cauchy problem has a unique local solution. With the help of the potential well argument, we characterize the stable sets by the asymptotic behavior of solutions as t goes to infinity, as well as the unstable sets by the blow-up of solutions in finite time.enAsymptotic behaviorFractional LaplacianStable and unstable setsWave equationsArticle