An inverse problem for identification of a time- and space-dependent memory kernel in viscoelasticity (Q2709866)

The authors are concerned with the problem of recovering the kernel \(m\) in the following second- order integrodifferential equation related to the rectangle \(\Omega=(0,1)\times {\mathbb{R}}_+\): NEWLINE\[NEWLINE \rho(x)D^2_tu(x,t) - D_x[\beta(x)D_xu(x,t)] + \int_0^t m(t-s)D_x[\beta(x)D_xu(x,s)] ds = f(x,t),\quad (x,t)\in \Omega,\tag{1}NEWLINE\]NEWLINE subject to the initial conditions NEWLINE\[NEWLINE u(x,0) = \varphi(x),\qquad D_tu(x,0) = \psi(x),\qquad x\in (0,1), \tag{2}NEWLINE\]NEWLINE and to the boundary conditions NEWLINE\[NEWLINE u(x,0) = f_1(t), \qquad u(x,1) = f_2(t), \quad t\in [0,+\infty). \tag{3}NEWLINE\]NEWLINE Functions \(\varphi\) and \(\psi\) are smooth, while \(\rho\), \(\beta\) and \(f_1\), \(f_2\) are, respectively, continuous on \([0,1]\) and \([0,+\infty)\). Moreover, \(\rho\) and \(\beta\) are positive on \([0,1]\). NEWLINENEWLINENEWLINEFinally, to recover \(m\), the following measurements on the state function \(u\) are prescribed: NEWLINE\[NEWLINE \Psi[u(\cdot,t)]=\beta(x_i)D_xu(x_i,t) -\int_0^t m(t-s)D_xu(x_i,s) ds=h_i(t), \quad t\in [0,T],\;i=1,\ldots, N, \tag{4}NEWLINE\]NEWLINE where \(x_i\) and \(h_i\) are, respectively, \(N\) observation points and \(N\) given bounded continuous functions. NEWLINENEWLINENEWLINE\noindent Using Laplace transform techniques the authors reduce the problem (1)-(4) to a fixed-point system for \(M\), the Laplace transform of \(m\), and solve in a specific space of analytic functions under suitable assumptions on the Laplace transforms \((f\beta){\widetilde {}}\), \({\widetilde \varphi}\), \({\widetilde \psi}\), and \({\widetilde h}_i\) of the data. Then such conditions are replaced by explicit (sufficient) conditions on the data \(f\), \(\varphi\), \(\psi\), and \(h_i\), \(i=1,\ldots, N\).