On determination of a time-dependent leading coefficient in a parabolic equation
The author considers the equation \[ u_t=a(t)u_{xx} + b(x,t)u_x +c(x,t)u + f(x,t),\quad (x,t)\in Q_{T}=(0,h)\times (0,T). \] Two inverse problems are studied. The first is to find a function \(a(t)\) and a solution to this equation that satisfies the conditions of the first initial-boundary value problem, \[ u(x,0)=\varphi(x), \;x\in [0,h],\quad u(0,t)=\mu_1(t),\;t\in [0,T], \] \[ u(h,t)=\mu_2(t), \;t\in [0,T]\tag{1} \] and the additional condition \[ a(t)u_x(0,t)=\mu_3(t), \;t\in [0,T]. \tag{2} \] The second problem differs from the first by the boundary conditions (1) and (2) which are replaced by the conditions \[ u_x(h,t)=\mu_2(t),\;\;u_x(0,t)=\mu_3(t),\;\;t\in [0,T]. \] Under appropriate conditions on the coefficients and the data of the problems, it is established that there exists a unique local (in time) solution \((a(t),u(x,t))\) to these problems which belongs to the Hölder class \(H^{1+\alpha/2}[0,t_0]\times H^{2+\alpha,1+\alpha/2}(\overline{Q_{t_0}})\). As a whole, the author's results are similar to the known results for the case \(b\equiv c\equiv 0\).
scientific article; zbMATH DE number 1225829