Identification of a special class of memory kernels in one-dimensional heat flow (Q2777924)

The authors are concerned with the problem of recovering the kernel \(m\) in the following integro-differential equation related the half-strip \(\Omega=(0,1)\times {\mathbb R}_+\): NEWLINE\[NEWLINE \beta(x)D_tu(x,t) - D_x(\lambda(x)D_xu)(x,t) NEWLINE\]NEWLINE NEWLINE\[NEWLINE + D_x\Big( \int_0^t m(t-s,x)D_x(x,s) ds\Big) = f(x,t),\qquad (x,t)\in \Omega, \tag{1}NEWLINE\]NEWLINE subject to the initial and boundary conditions NEWLINE\[NEWLINE u(x,0) = \varphi(x),\qquad x\in [0,1], \tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0,t) = f_1(t),\qquad u(0,t) = f_2(t),\qquad t\in [0,+\infty).\tag{3}NEWLINE\]NEWLINE Functions \(\beta\), \(\lambda\), \(\varphi\), \(f\), \(f_1\) and \(f_2\) are smooth and, in addition, \(\beta\) and \(\lambda\) are positive on \([0,1]\). NEWLINENEWLINENEWLINE\noindent Moreover, the authors assume that the unknown kernel \(m\) is degenerate, i.e. NEWLINE\[NEWLINE m(x,t) = \sum_{k=1}^n m_k(t)\mu_k(x), \tag{4}NEWLINE\]NEWLINE where the \(\mu_k\)'s, \(k=1,\dots,N,\) are given (smooth) functions on \([0,1]\) and the \(m_k\)'s, \(k=1,\dots,N,\) are unknown memory kernels on \([0,+\infty)\). NEWLINENEWLINENEWLINEThe \(N\) additional conditions needed to recover functions \(m_k\) are of the following forms, where \(h_i\), \(i=1,\dots,N,\) are given (smooth) functions: NEWLINE\[NEWLINE u(x_i,t)=h_i(t),\qquad t\in [0,+\infty),\;i=1,\dots,N_1,\qquad 0\leq N_1\leq N, \tag{5}NEWLINE\]NEWLINE NEWLINE\[NEWLINE -\lambda(x_i)D_xu(x_i,t) +\int_0^t m(x_i,t-s)D_xu(x_i,s)ds=h_i(t), NEWLINE\]NEWLINE NEWLINE\[NEWLINE t\in [0,+\infty),\;i=N_1+1,\dots,N. \tag{6}NEWLINE\]NEWLINE The observation points \(x_i\) belong, respectively, to \((0,1)\), if \(i=1,\dots,N_1,\) and to \([0,1]\), if \(i=N_1+1,\dots,N.\) NEWLINENEWLINENEWLINEUsing Laplace transforms, the authors reduce problem (1)--(6) to a fixed-point system for \((M_1,\dots M_N)\), the Laplace transform of \((m_1,\dots m_N)\). Taking advantage of specific regularity properties of the one-dimensional Green function related to the transformed problem, the authors can solve the previous system in a specific space of analytic functions and prove that problem (1)--(6) has a unique solution \((u,m)\) for all positive \(t\), provided an explicit operator of the data admits a suitable representation.