Global behavior of the energy to the hyperbolic equation of viscoelasticity with combined power-type nonlinearities
Haiyang Lin, Jinqi Yan, Bo You, Marcelo M. Cavalcanti
TL;DR
This work analyzes the global dynamics of viscoelastic wave equations with full-history memory, nonlinear damping, and a supercritical, combined power-type source. Using a weighted potential well framework, energy identities, and careful blow-up analysis, it establishes global existence under energy-dissipation-dominant or potential-well conditions, and derives explicit energy-decay rates that reflect the relaxation kernel and damping growth. It also provides a comprehensive blow-up theory, showing finite-time blow-up for negative and certain nonnegative initial energies, and even blow-up for arbitrary initial energy through constructive initial data. The results offer a rigorous understanding of stability versus instability in viscoelastic media with memory effects and complex nonlinearities, with implications for materials exhibiting hereditary behavior under dynamic loading.
Abstract
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.