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GLOBAL NONEXISTENCE OF THE SOLUTIONS FOR A NONLINEAR WAVE EQUATION WITH THE Q-LAPLACIAN OPERATOR

We study the global nonexistence of the solutions of the nonlinear q-Laplacian wave equation u tt − ∆ q u + (−∆) α u t = |u| p−2 u, where 0 < α ≤ 1, 2 ≤ q < p. We obtain that the solution blows up in finite time if the initial energy is negative. Meanwhile, we also get the solution blows up in finite time with suitable positive initial energy under some conditions.