Identifying a space dependent coefficient in a reaction-diffusion equation

We consider a reaction-difiusion equation for the front motion u in which the reaction term is given by c(x)g(u). We formulate a suitable inverse problem for the unknowns u and c, where u satisfies homogeneous Neumann boundary conditions and the additional condition is of integral type on the time interval [0; T]. Uniqueness of the solution is proved in the case of a linear g. Assuming g non linear, we show uniqueness for large T.