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One spatial variable thermoelastic transmission problem in viscoelasticity located in the second part. (English) Zbl 1416.35159

The authors deal with a thermoelastic transmission problem \[ \begin{aligned}[t] &\rho_1u''-a_1u_{xx}+\beta_1\theta_x=0,\ c_1w_1''-\ell\theta_{xx}+\beta_1u_x'=0,\, x\in (-L,0),\\ &\rho_2v''-a_2\left(v_{xx}-\int_{-\infty}^t\mu(t-s)v_{xx}(s)ds \right)+\beta_2q_x=0,\\ & c_2w_2''-kw_{2,xx}+\beta_2v_x'=0,\, x\in (0,L),\, t>0,\\ &u(-L,t)=v(L,t)=w_1(-L,t)=w_2(L,t)=0,\\ &u(0,t)=v(0,t),\ \theta(0,t)=q(0,t),\ w_1(0,t)=w_2(0,t),\ \ell\theta_x(0,t)=kw_{2,x}(0,t),\\ &a_1u_x(0,t)-a_2v_x(0,t)=\beta_1\theta(0,t)+\beta_2 q(0,t),\ t>0,\\ &u(\cdot,0)=u^0(x),\ u'(\cdot,0)=u^1(x),\ w_1(\cdot,0)=w_1^0(x),\ \theta(\cdot,0)=\theta^0(x),\ x\in (-L,0),\\ &v(\cdot,0)=v^0(x),\ v'(\cdot,0)=v^1(x),\ w_2(\cdot,0)=w_2^0(x),\ q(\cdot,0)=q^0(x),\ x\in (0,L), \end{aligned} \] where \(u,\,v\) are displacements of the system and \(\theta,\,q\) are the temperature differences with respect to a fixed reference temperature in \((-L,0)\) and \((0,L)\), respectively. The thermal displacements \(w_1,\,w_2\) satisfy the relations \[ w_1(\cdot,t)=\int_0^t \theta(\cdot,s)ds+w_1(\cdot,0),\quad w_2(\cdot,t)=\int_0^t \theta(\cdot,s)ds+w_2(\cdot,0). \] The main result is the dissipation of the system with the \(t^{-1}\) sharp decay rate, and, hence, the dissipation produced by the viscoelastic part is not strong enough to produce an exponential decay of the solution.

MSC:

35L53 Initial-boundary value problems for second-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
93D20 Asymptotic stability in control theory
76B75 Flow control and optimization for incompressible inviscid fluids
35R09 Integro-partial differential equations
74F05 Thermal effects in solid mechanics
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