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Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. (English) Zbl 1188.35029

The authors deal with the initial-boundary value problem for a system of viscoelastic wave equations in a form

u tt (x,t)-Δu+ 0 t g(t-s)Δu(x,s)ds+|u t | m-1 u t =f 1 (u,v),v tt (x,t)-Δv+ 0 t h(t-s)Δv(x,s)ds+|v t | r-1 v t =f 2 (u,v),xΩ,t>0,u(x,t)=v(x,t)=0,xΩ,t0,u(0),v(0)=(u 0 ,v 0 ),u t (0),v t (0)=(u 1 ,v 1 ),xΩ,

where Ω is a bounded domain of N (N1) with a smooth boundary Ω. They prove a global nonexistence of solutions for a large class of initial data for which the initial energy takes positive values.

MSC:
35B44Blow-up (PDE)
35B40Asymptotic behavior of solutions of PDE
35L71Semilinear second-order hyperbolic equations
35L20Second order hyperbolic equations, boundary value problems
35R09Integro-partial differential equations
74D05Linear constitutive equations (materials with memory)
35L53Second-order hyperbolic systems, initial-boundary value problems
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