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: ${{u}_{tt}}-{{\Delta }_{m}}u-\Delta u+g*\Delta u-{{\mu }_{1}}\Delta {{u}_{t}}\left( x, t \right)-{{\mu }_{2}}\Delta {{u}_{t}}\left( x, t-\tau \right)={{\left| u \right|}^{p-2}}u$. When the initial energy 0 < E(0) < E₁, the concave function L₁(t) was constructed with the energy function, and the differential inequality $\frac{\text{d}{{L}_{1}}\left( t \right)}{\text{d}t}\ge {{\xi }_{0}}L_{1}^{1+\nu }\left( t \right)\ \left( {{\xi }_{0}}>0, \nu >0, t\ge 0 \right)$ 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.