Abstract
In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if p > m, and a global result for p ≤ m.
Highlights
A purely elastic material has a capacity to store mechanical energy with no dissipation
The important thing about viscous materials is that when the force is removed it does not return to its original shape
Materials which are outside the scope of these two theories will be those for which some, but not all, of the work done to deform them can be recovered. Such materials possess a capacity of storage and dissipation of mechanical energy
Summary
A purely elastic material has a capacity to store mechanical energy with no dissipation (of the energy). He proved a blow up result for the solution with negative initial energy if p > m , and a global result for p ≤ m. The case of linear damping (m = 2) and nonlinear source has been first considered by Levine [7] [8] He showed that solutions with negative initial energy blew up in finite time. Where Ω was a bounded domain of Rn (n ≥ 1) with a smooth boundary ∂Ω , m ≥ 2 , p > 2 , and g : R+ → R+ was a positive nonincreasing function They showed, under suitable conditions on g , that there were solutions of (1.5) with arbitrarily high initial energy that blow up in a finite time. C denotes a general positive constant, which may be different in different estimates
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