Asymptotic behaviour of solutions of an inverse problem for parabolic equations with integral overdetermination (Q2719455)

In this paper the authors consider a linear inverse problem of defining the couple of functions \(\{u(t,x),f(t)\}\) for the equations NEWLINE\[NEWLINEu_t-\Delta u+\alpha u=f(t)g(t,x)+h_0(t,x) + \sum_{j=1}^n \frac{\partial}{\partial x_j} h_j(t,x),\quad (t,x)\in Q_\infty,NEWLINE\]NEWLINE with the initial condition \(u(0,x) = u_0(x)\), \(x\in\Omega\), the boundary condition \(u(t,x) = 0\), \(x\in\partial\Omega\), \(0\leq t<+\infty\), and the condition of integral redefinition NEWLINE\[NEWLINE \int_\Omega u(t,x)w(x) dx = \varphi(t),\quad 0\leq t<+\infty.NEWLINE\]NEWLINE Here \(Q_\infty = \{(t,x): 0<t<+\infty\), \(x\in\Omega\}\), \(\Omega\subset \mathbb{R}^n\) is the bounded domain with smooth boundary, the functions \(g\), \(u_0\), \(w\), \(\varphi\), \(h_j\), \(j=1,2,\dots,n\) and the constant \(\alpha\) are given. The authors prove theorems on stabilization to zero of the solution of an inverse problem for parabolic equation and on asymptotic closeness of solutions of inverse problems for parabolic and elliptic equations.