Blow-up of Solutions to the m-Laplacian Type Wave Equation with Strong Delay Terms

Blow-up of solutions to the m-Laplacian type wave equation with strong delay was studied: u_tt-\Delta _mu-\Delta u+g*\Delta u-\mu _1\Delta u_t\left( x, t \right)-\mu _2\Delta u_t\left( x, t-\tau \right)=\left| u \right|^p-2u. When the initial energy 0 < E(0) < E₁, the concave function L₁(t) was constructed with the energy function, and the differential inequality \frac\textdL_1\left( t \right)\textdt\ge \xi _0L_1^1+\nu \left( t \right)\ \left( \xi _0>0, \nu >0, t\ge 0 \right) was obtained. Then, the differential inequality was integrated in (0, t), and it was proved that there was a finite time T^*>0, so that when the time t was tended to T^*, the m-Laplacian type wave equation underwent blow-up of solutions. When the initial energy E(0) < 0, a concave function L₂(t) was also constructed. With the same method, it was found that the solutions to the equation had a finite-time blow-up.