Blow-up phenomena for a nonlinear pseudo-parabolic equation with nonlocal source

Abstract

We discuss a class of quasi-linear pseudo-parabolic equation with nonlocal source u t − Δ u t − ∇ ⋅ | ∇ u | 2 q ∇ u = u p ( x , t ) ∫ Ω k ( x , y ) u p + 1 ( y , t ) d y x , y ∈ Ω , t ∈ ( 0 , T 0 ] , where q ≤ p and 0 < q < n − 2 2 . By establishing the criterions for blow-up, we determine the upper bounds for blow-up time under not only q < p and non-positive initial energy but also q = p and negative initial energy. The results show that the upper bound for blow-up time under q < p is different from it under q = p . Moreover, we also determine the lower bound for blow-up time.