Determination of an unknown function in parabolic problems with standard and non-standard boundary data (Q1181068)

The following identification problem of one-dimensional heat conduction is considered: Determine a function \(P(z)\) defined on \([0,M]\) for some \(M\) and a function \(u(x,t)\) to satisfy the system \[ u_ t=\alpha u_{xx}-\beta u_ x+\gamma u+q(x,t) \quad\text{in}\quad \Omega_ T=\{(x,t):\;0<x<1,\;0<t<T\},\tag{1} \] \[ u(x,0)=f(x), \qquad 0<x<1,\tag{2} \] \[ u_ t(0,t)-u_ x(0,t)+P(u(0,t))=g(t), \qquad 0<t<T,\tag{3} \] \[ u_ x(1,t)=h(t), \qquad 0<t<T,\tag{4} \] together with the overspecified boundary condition \[ u(0,t)=k(t), \qquad 0<t<T.\tag{5} \] It is shown that, using the Green's function for the operator \(\partial/\partial t- \alpha\partial^ 2/\partial x^ 2\) in the domain \(\Omega_ T\), the combination \(u_ t(0,T)-u_ x(0,t)\) can be represented as a function of \(u(0,t)\). Two more versions of the basic problem (1)--(5) are analyzed in which the boundary condition (4) or (5) is replaced by any of the non-standard conditions \[ \int_ 0^ 1 \exp[-\beta x/2\alpha]u(x,t)dx=\varphi(t), \qquad \int_ 0^ t \eta(t-\tau)u(0,\tau)d\tau=\psi(t), \qquad 0<t<T. \] Under certain restrictions on the initial and boundary data, the proposed identification problems are proven to be well-posed in the classical sense.