On some linear inverse heat conduction problems (Q1893927)

The author considers inverse problems of finding one of the coefficients \(a_i(x)\) \((i = 1, \ldots, 4)\) as well as the function \(u(x,t)\) satisfying the system \[ u_t = \bigl( a_4 u_x \bigr)_x - a_3 u_x - a_2 u\;- a_1 f, \quad 0 < x < 1, \quad 0 < t \leq T,\tag{1} \] \[ (2) \quad u(0,t) = g(t), \quad (3) \quad u_x(0,t) = 0\;(0 < t \leq T), \quad (4) \quad u(x,0) = u_0(x)\;(0 < x < 1). \] The solution \(u(x,t)\) is defined in a weak sense and \(a_i(x)\) \((i = 1, \dots, 4)\) are sought in \(L^\infty (0,1)\). The following results are contained in the paper. 1. If some additional conditions are fulfilled, then the inverse problem of determining a pair of functions \(\{a_1(x), u(x,t)\}\) has a unique solution, and the inverse problem of determining a pair of functions \(\{a_j(x), u(x,t)\}\) \((j = 2,3,4)\) admits at most one solution. 2. The solvability of the problem (1)--(3), where the coefficients \(a_i\) \((i = 1, \ldots, 4)\) are not only functions of \(x\) but of the variables \(x\) and \(t\) is investigated. It is shown that for the noncharacteristic Cauchy problem an initial condition (4) at \(t = 0\) is not necessary.