Heat equation with an unknown diffusion coefficient
Not always. Suppose the boundary conditions are
$$ u(0,t) = u(L,t) = T_0 $$
Then $u \equiv T_0$ is a solution to the heat equation for any diffusion coefficient $k(x)$.
Not always. Suppose the boundary conditions are
$$ u(0,t) = u(L,t) = T_0 $$
Then $u \equiv T_0$ is a solution to the heat equation for any diffusion coefficient $k(x)$.