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Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. (English) Zbl 1098.35031

The nonlinear viscoelastic equation with damping and source terms,

u tt -Δu+ 0 t g(t-τ)δu(τ)dτ+u t |u t | m-2 =u|u| p-2

with initial conditions and Dirichlet boundary conditions is considered. The existence of local solutins is proved. For nonincreasing positive g and for p>max(2,m),max(m,p)2(n-1) (n-2),n3,m>1 the author proves that there are solutions with positive initial energy that blow up in finite time.

35B40Asymptotic behavior of solutions of PDE
45K05Integro-partial differential equations
35Q72Other PDE from mechanics (MSC2000)
74B20Nonlinear elasticity