The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation:which has been studied in [5]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the L 2 (Ω) norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [5].

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This paper deals with the lower bound for blow-up solutions to a quasilinear hyperbolic equation with strong damping. An inverse Hölder inequality with a correction constant is employed to overcome the difficulty caused by the failure of the embedding inequality. Moreover, a lower bound for blow-up time is obtained by constructing a new control functional with a small dissipative term and by applying an inverse Hölder inequality as well as energy inequalities. This result gives a positive answer to the open problem presented in [1].