Asymptotic expansion of solutions to an inverse problem of parabolic type (Q955876)

The author is concerned with the stabilization and the asymptotic expansion of the solution to a first-order in time equation related to a Banach space \(X\) when the source function is unknown and an additional information is supplied. More exactly, he wants to determine two functions \(u:[S,+\infty)\to X\) and \(f:[S,+\infty)\to\mathbb R\), \(S\in \mathbb R\), solving the following identification problem depending on the complex parameter \(\lambda\): \[ u'(t)=[A(t)-\lambda]u(t)+f(t)z(t)+h(t),\quad t\in [S,+\infty), \tag{1} \] \[ u(S)=u_S, \tag{2} \] \[ \Phi[u(t)]=g(t),\quad t\in [S,+\infty). \tag{3} \] The operator family \(\{A(t)\}_{t\in [S,+\infty)}\) is assumed to consist of sectorial operators with a fixed domain \(D\) and \(\Phi\) is assumed to be a linear continuous functional on \(X\), while the reference space for equation (1) is \(X_\theta\), an interpolation space between \(D\) and \(X\). The function \(u\) is looked for in the Sobolev space \(W^{2,1}((S,+\infty);X_\theta)\) to insure the existence of the limit of \(u\) and \(u'\) as \(t\to +\infty\). The first result guarantees that the solution \((u,f)\) exists in \([S,+\infty)\) and has a limit \((u_0,f_0)\in X\times \mathbb R\) as \(t\to +\infty\), provided that \(A(t)\), \(z(t)\), \(h(t)\), \(g(t)\) satisfy suitable assumptions and have limits \(A_0\), \(z_0\), \(h_0\), \(g_0\) in suitable spaces as \(t\to +\infty\), with \(\Phi[z_0]\neq 0\) and the resolvent set of the auxiliary operator \(B_0=A_0-\lambda - \Phi[z_0]^{-1}\Phi[A_0\cdot]z_0\) contains the half-plane \(\text{Re}\,\mu\geq 0\). Then, under the assumptions that \(A(t)\), \(z(t)\), \(h(t)\), \(g(t)\) admit a finite expansion, as functions of \(t^{-1}\), as \(t\to +\infty\), the author can prove that also \((u(t),f(t))\) has a similar expansion. Finally, the previous abstract results are applied to a similar identification problem for general linear parabolic equations of order \(2m\).