Identification of weakly singular relaxation kernels in three-dimensional viscoelasticity (Q5952259)

The author is concerned with the following operator identification problem related to a Hilbert space \(X\), a Banach space \(Y\hookrightarrow X\) (densely) and to a closed subspace \(K\subset {\mathcal L}(X)\): determine a pair of functions \(u:(0,+\infty)\to X\) and \(M:(0,+\infty)\to K\) satisfying the equations NEWLINE\[NEWLINE u''(t) = Au(t) - \int_0^t M(t-s)Bu(s) ds + f(t),\qquad t\in (0,+\infty), \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0) = \varphi,\qquad u'(0) = \psi, \tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \Psi u(t) = H(t),\qquad \forall t\in (0,+\infty), \tag{3}NEWLINE\]NEWLINE where \(f\), \(\varphi\), \(\psi\) and \(H\) are smooth given functions. Moreover, the author assumes that \(A\) is a linear closed operator densely defined in \(X\) with domain \({\mathcal D}(A)=Y\) and non-empty resolvent set, while \(B\) and \(\Psi\) are elements in \({\mathcal L}(Y;X)\) and \({\mathcal L}(X;K)\), respectively. NEWLINENEWLINENEWLINEThe unknown operator \(M\) is assumed to be not locally differentiable so that \(A\) is \textit{not} dominant with respect to the integral term. Consequently, to recover \(M\), the author must assume that it admits some special decomposition \(M=M_0+M_1\), \(M_0\) and \(M_1\) belonging, respectively, to a set of monotone (possibly singular) operators and to a set of smooth operators. NEWLINENEWLINENEWLINEUnder suitable assumptions on the data the author shows that that problem (1)--(3) admits a unique solution defined in \((0,+\infty)\). NEWLINENEWLINENEWLINE\noindent Finally, the abstract result is applied to the explicit problem consisting of recovering the two scalar kernels \(M^1,M^2:(0,+\infty)\to {\mathbb R}\) in the following identification vector problem, related to a bounded smooth domain \(\Omega\): NEWLINE\[NEWLINE\begin{multlined} D_t^2u_i(x,t) = (g^1+g^2)\sum_{j=1}^3 D_{x_i}D_{x_j}u_j(x,t) + g^2\Delta u_i(x,t)\\ - \int_0^t [M^1(t-s)+M^2(t-s)]\sum_{j=1}^3 D_{x_i}D_{x_j}u_j(x,s)\\ + M^2(t-s)\Delta u_i(x,t) ds + f_i(x,t),\quad (x,t)\in \Omega \times (0,+\infty)\end{multlined} NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0) = \varphi(x),\quad D_tu(x,0) = \psi(x),\quad \forall x\in \Omega, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u_i(x,t) = 0,\quad \forall (x,t)\in \partial \Omega \times [0,+\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINE \int_{\Gamma^\nu} \sum_{i,j=1}^3 n_i(x)n_j(x)D_{x_j}u_i(x,t) d\sigma(x) = h^\nu,\quad t\in (0,+\infty),\;\nu=1,2 NEWLINE\]NEWLINE where \(n(x)\) denotes the outward unit vector at \(x\in \partial \Omega\) and \(\Gamma^\nu\), \(\nu=1,2\), are subsets of \(\partial \Omega\) with positive surface measure such that their intersection has a null measure.