Exponential decay of energy of vibrating strings with local viscoelasticity (Q1601104)

The authors study the energy decay rate for an one-dimensional linear wave equation with a viscoelastic damping acting on a subinterval. It models an string with one segment made of viscoelastic material and the other of elastic material. They distinguish the short memory Kelvin-Voigt damping and the long memory Boltzmann model. The initial-boundary value problem corresponding to the Kelvin-Voigt model is \[ m(x)\ddot u(x,t)=[a(x)u'(x,t)+b(x)\dot u'(x,t)]',\quad x\in (0,L),\;t>0, \] \[ u(0,t)=u(L,t)=0,\quad u(x,0)=u^0(x),\quad \dot u(x,0)=u^1(x), \] and to the Boltzmann model \[ \ddot u(x,t)=\left[a(x)u'(x,t)-b(x)\int_0^\infty g_s(s)(u'(x,t)-u'(x,t-s)) ds\right]', \] \[ u(0,t)=u(L,t)=0,\;t>0,\;u(x,t)=u^0(x,t),\;\dot u(x,t)=u^1(x,t),\;t\leq 0,\;0<x<L. \] The localization of dampings is obtained by assuming that \(b(x)\) has a support \((\alpha,\beta) \subset (0,L)\). In the case of the Kelvin-Voigt model the conditions on the coefficients are \(m,a,b\in C^1[0,L]\), \((ma)'\), \((ab')\in C^{0,1}[0,L]\), \(m(x)>0\),\(a(x)>0\), \(b(x)\geq 0\), \(\int_0^L b(x) dx\geq 0\). The corresponding semigroup \(e^{\mathcal A} t\) is then exponentially stable and hence the energy \(E(t)=\frac 12 \int_0^T [m(x)|\dot u|^2+a(x)|u'|^2] dx\) decays exponentially. The same results are achieved in the case of the Boltzmann model with coefficients fulfilling the conditions \(g(s)\in C^2(0,\infty)\cap C[0,\infty)\), \(g_s(s)\in L^1(0,\infty)\), \(g>0\), \(g_s<0\), \(g_{ss}>0\) on \((0,\infty),\) \(-kg_s\leq g_{ss}\leq -Kg_s\) on \((0,\infty)\), \(0<k<K\), \(g(\infty)=0,\) \(a,b\in C^{0,1}[0,\alpha]\cap C^{0,1}[\alpha,L]\), \(\alpha>0\), \(b>0\) on \((\alpha,L]\), \(b=0\) on \([0,\alpha),\) \(a+g(0)b\in C^{1,1}[\alpha,L]\), if \(b(\alpha^+)=0.\)