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
\[ \begin{aligned} &u_{tt}(x,t)- \Delta u+ \int_0^tg(t-s)\Delta u(x,s)\,ds+ | u_t|^{m-1}u_t= f_1(u,v),\\ &v_{tt}(x,t)- \Delta v+ \int_0^th(t-s)\Delta v(x,s)\,ds+ | v_t|^{r-1}v_t= f_2(u,v),\quad x\in \Omega,\;t>0,\\ &u(x,t)=v(x,t)=0,\quad x\in \partial\Omega,\;t\geq 0,\\ &\left(u(0),v(0)\right)=(u_0,v_0),\;\left(u_t(0),v_t(0)\right)=(u_1,v_1),\quad x\in \Omega,\end{aligned} \]
where \(\Omega\) is a bounded domain of \(\mathbb R^N\) \((N\geq 1)\) 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.
\[ \begin{aligned} &u_{tt}(x,t)- \Delta u+ \int_0^tg(t-s)\Delta u(x,s)\,ds+ | u_t|^{m-1}u_t= f_1(u,v),\\ &v_{tt}(x,t)- \Delta v+ \int_0^th(t-s)\Delta v(x,s)\,ds+ | v_t|^{r-1}v_t= f_2(u,v),\quad x\in \Omega,\;t>0,\\ &u(x,t)=v(x,t)=0,\quad x\in \partial\Omega,\;t\geq 0,\\ &\left(u(0),v(0)\right)=(u_0,v_0),\;\left(u_t(0),v_t(0)\right)=(u_1,v_1),\quad x\in \Omega,\end{aligned} \]
where \(\Omega\) is a bounded domain of \(\mathbb R^N\) \((N\geq 1)\) 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.
Reviewer: Igor Bock (Bratislava)
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35R09 | Integro-partial differential equations |
| 74D05 | Linear constitutive equations for materials with memory |
| 35L53 | Initial-boundary value problems for second-order hyperbolic systems |
Keywords:
blow up; global nonexistence; positive initial energy; viscoelastic wave equations; initial-boundary value problemReferences:
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