Paper detail

Finite time blow-up and global solutions for the viscoelastic wave equation with combined power-type nonlinearities

The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a supercritical source term which is a combined power-type nonlinearities. The global existence of the solutions is obtained provided that the energy sink dominates the energy source in an appropriate sense. In more general scenarios, we prove the global existence of the solutions if the initial history value $u_0$ is taken from a subset of a suitable potential well. Based on global existence results, the energy decay rate is derived which depends on the relaxation kernel as well as the growth rate of the damping term. In addition, we study blow-up of solutions when the source is stronger than dissipation.