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Zbl 1188.35029
Messaoudi, Salim A.; Said-Houari, Belkacem
Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. (English)
[J] J. Math. Anal. Appl. 365, No. 1, 277-287 (2010). ISSN 0022-247X

The authors deal with the initial-boundary value problem for a system of viscoelastic wave equations in a form $$\align &u_{tt}(x,t)- \Delta u+ \int_0^tg(t-s)\Delta u(x,s)\,ds+ \vert u_t\vert^{m-1}u_t= f_1(u,v),\\ &v_{tt}(x,t)- \Delta v+ \int_0^th(t-s)\Delta v(x,s)\,ds+ \vert v_t\vert^{r-1}v_t= f_2(u,v),\quad x\in \Omega,\ t>0,\\ &u(x,t)=v(x,t)=0,\quad x\in \partial\Omega,\ t\ge 0,\\ &\left(u(0),v(0)\right)=(u_0,v_0),\ \left(u_t(0),v_t(0)\right)=(u_1,v_1),\quad x\in \Omega,\endalign$$ where $\Omega$ is a bounded domain of $\Bbb R^N$ $(N\ge 1)$ with a smooth boundary $\partial\Omega$. They prove a global nonexistence of solutions for a large class of initial data for which the initial energy takes positive values.
[Igor Bock (Bratislava)]

MSC 2000:: *35B44
35B40 Asymptotic behavior of solutions of PDE
35L71
35L20 Second order hyperbolic equations, boundary value problems
35R09
74D05 Linear constitutive equations
35L53

Keywords: blow up; global nonexistence; positive initial energy; viscoelastic wave equations; initial-boundary value problem

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