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Global solutions to quasi-linear hyperbolic systems of viscoelasticity. (English) Zbl 1228.35136

The authors deal with the initial value problem
\[ \begin{aligned} &u_{tt}-\sum_{j=1}^n b^j(\partial_xu)_{x_j}+\sum_{j,k=1}^n K^{jk}*u_{x_jx_k}+Lu_t=0,\\ &u(x,0)=u_0(x),\;\;u_t(x,0)=u_1(x),\quad x\in \mathbb R^n \end{aligned} \]
with smooth \(m\)-vector functions \(b^j\), an unknown \(m\)-vector function \(u\), smooth \(m \times m\) real matrix functions \(K^{jk}\) and the symmetric \(m\times m\) matrix \(L\). The system is assumed to have a free energy \(\varphi(v)\) satisfying \(b^j(v)=D_{v_j}\varphi(v)\), where \(D_{v_j}\varphi(v)\) denotes the Fréchet derivative of \(\varphi(v)\) with respect to \(v_j\). The system has dissipative properties due to a memory and a damping term. It is proved that the solution exists globally in time, provided that the initial data are sufficiently small. Moreover the solution converges to zero as time tends to infinity. The crucial point of the proof are uniform a priori estimates of solutions by using an energy method.

MSC:

35L52 Initial value problems for second-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
35L72 Second-order quasilinear hyperbolic equations
35R09 Integro-partial differential equations
74D10 Nonlinear constitutive equations for materials with memory
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References:

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