Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level. Issue 12 (10th September 2019) - Sun, Fenglong; Liu, Lishan; Wu, Yonghong
ABSTRACT: In this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history u 0 ( x, t ), t ≤ 0 : u tt - k ( 0 ) Δ u - ∫ 0 + ∞ k ′ ( s ) Δ u ( t - s ) d s + | u t | m - 1 u t = | u | p - 1 u, in Ω × ( 0, T ), where the relaxation function k is monotone decreasing with k ( + ∞ ) = 1, m ≥ 1 and 1 ≤ p < 6 . When the source is stronger than dissipations, i.e. p > max m, k ( 0 ) , we obtain some finite time blow-up results with positive initial energy. In particular, we obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.