Lifespan for a semilinear pseudo‐parabolic equation

This paper deals with the blow‐up solution to the following semilinear pseudo‐parabolic equationu t = Δ u t + Δ u + | u | p − 1 u in a bounded domain Ω ⊂ R n , which was studied by Luo (Math Method Appl Sci 38(12):2636‐2641, 2015) with the following assumptions on p :1 < p < + ∞ ,if n = 1 , 2 ;1 < p ≤ n + 2 n − 2 ,if n ≥ 3 ,and the lifespan for the initial energy J ( u 0 )<0 is considered. This paper generalizes the above results on the following two aspects: a new blow‐up condition is given, which holds for all p> 1; a new lifespan is given, which holds for all p> 1 and possible J ( u 0 )≥0. Moreover, as a byproduct, we refine the lifespan when J ( u 0 )<0.