On an inverse problem for quasilinear parabolic equations (Q1281867)

Under specific conditions the problem of identifying the parameter function \(a(x,u)\) in the initial-boundary value problem \[ u_t- a(x,u) u_{xx}= 0,\quad 0<x<1,\quad 0<t<T, \] \[ u(x,0)= 0,\quad 0<x<1, \] \[ u(0,t)= f(t),\quad u(1,t)= 0,\quad 0< t<T \] is under consideration in this paper. In particular, uniqueness results for \(a(x,u)\) based on the Dirichlet-to-Neumann map \[ \Lambda(a,f): u(0,t)= f(t)\mapsto u_x(0,t)\quad\text{on }[0, T] \] are of interest. For a subclass of admissible parameter functions it follows that \(\Lambda(a^1, f)= \Lambda(a^2, f)\) implies \(a^1= a^2\) on \([0,1]\times [0,f(T)]\).