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Zbl 1098.35031
Messaoudi, Salim A.
Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. (English)
[J] J. Math. Anal. Appl. 320, No. 2, 902-915 (2006). ISSN 0022-247X

The nonlinear viscoelastic equation with damping and source terms, $$ u_{tt}-\Delta u + \int^t_0 g(t-\tau) \delta u(\tau)d\tau + u_t\vert u_t\vert^{m-2}=u\vert u\vert^{p-2} $$ with initial conditions and Dirichlet boundary conditions is considered. The existence of local solutins is proved. For nonincreasing positive $g$ and for $p> \max(2,m),\, \max(m,p)\leq \frac{2(n-1)}{(n-2)}, \, n \geq 3,\,m>1$ the author proves that there are solutions with positive initial energy that blow up in finite time.
[Nickolaj A. Lar'kin (Maringa)]

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
45K05 Integro-partial differential equations
35Q72 Other PDE from mechanics
74B20 Nonlinear elasticity

Keywords: blow-up; viscoelastic equations; nonlinear damping; positive initial energy; Dirichlet boundary conditions

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