An inverse problem for a parabolic partial differential equation
This is a very interesting problem and the paper seems to me to be a good piece of work. The investigation involves finding the solution u(x,t) and the coefficient a(x) in the boundary-initial value problem \[ (P)u_ t- u_{xx}+a(x)u=0,\quad 0<x<1,\quad 0<t<T,\quad u(0,t)=u(1,t)=0,\quad 0\leq t\leq T, \] \[ u(x,0)=f(x),\quad 0\leq x\leq 1,\quad u_ x(0,t)=g(t),\quad 0\leq t\leq T, \] where a condition like the last is necessary since u(x) is also unknown. Under various regularity assumptions on f and the requirements that \(g\in C^ 1[0,T],\) is analytic on the half plane \(Re(t)>0\) and possesses a Dirichlet series \(g(t)=\sum^{\infty}_{1}a_ n\exp(-\lambda_ nt)\) with \(\{\lambda_ n\}\) a sequence of positive, increasing real numbers with asymptotic behaviour \(n\sqrt{\lambda_ n}=n\Pi +O(1/n),\) it is shown that there exists a unique solution to (P). The proof is illustrative and the conditions on g are shown to be sharp. Finally, the question of the continuous dependence of a(x) on f(x) and g(t) is studied.
scientific article; zbMATH DE number 3849744