Recovering a time dependent coefficient in a parabolic differential equation (Q1192554)

The authors discuss the problem of finding the function \(u(x,t)\) and the unknown positive coefficient \(a(t)\) in the parabolic initial-boundary value problem \[ u_ t- a(t)\Delta u=0 \quad \text{in } \Omega\times (0,T], \] \[ u(x,t)=f(x,t) \quad \text{on }\partial\Omega \times[0,T], \qquad u(x,0)=h(x) \quad \text{in }\Omega, \] where \(\Omega\) is a simply connected domain in \(\mathbb{R}^ n\) with smooth boundary \(\partial\Omega\). With only the above data this problem is under-determined and the authors are forced to impose additional boundary conditions. They show that a unique solution pair \(\langle u,a\rangle\) is obtained when in addition one prescribes certain time dependent functions of \(u\). In particular, this may take the form of the heat flux \(g(t)\) at a given point \(x_ 0\in \partial\Omega\), that is, \[ -a(t) {{\partial u} \over {\partial\nu}} (x_ 0,t)= g(t), \qquad 0\leq t\leq T, \] where \(\nu\) denotes the inner normal to the curve \(\partial\Omega\). Alternatively one may impose other functionals, not necessarily those that correspond to measurements on the lateral boundary, for example, the total energy leaving the domain \(\Omega\) can be specified at each time \(t\) \[ \int_ \Omega u(x,t)dx= E(t), \qquad 0\leq t\leq T, \] and a recovery of the function \(a(t)\) can be made.