Consider the following initial-boundary value problem for the heat equation: $$u_t(x,t)=u_{xx}(x,t),\ \ \ x\in[0,1] \\ u(x,0)=u_0\\ a(t)u(0,t)+b(t)u'(0,t)=c(t)u(1,t)+d(t)u'(1,t)=0$$ Meaning, our homogeneous boundary conditions are time dependent. $a,b,c,d,u_0$ are all smooth functions.
In the time independent case, such a problem can be solved using separation of variables. The time dependent case I presented should be more complicated, but as a first step I tried to think if separated solutions could exist. I suggest a solution of the form $$u(x,t)=X(x)T(t)$$ Plugging this into the heat equation I get as usual $$XT'=X''T\Rightarrow\frac{T'}{T}=\frac{X''}{X}.$$ Since the LHS depends only on $t$ and the RHS depends only on $x$, we conclude that both sides should be equal to a constant $\lambda$. For each given $t$, this $\lambda$ should be an eigenvalue of the Laplacian on $[0,1]$ equipped with the boundary conditions corresponding to time $t$. But since the boundary conditions are time dependent, the eigenvalues of the Laplacian will be time dependent as well, which means that in fact $\lambda$ must be a function of $t$ and not a constant (except for some very degenerate cases).
Does this mean that if the (homogeneous) boundary conditions are time dependent (and the eigenvalues of the Laplacian change in time), then the heat equation can never have any separated solutions?
More generally, is there another way to solve problems of this type, since separation of variables does not seem to work? Are there known cases where it is possible to solve such problem using something similar to separation of variables? I prefer not to use perturbative approaches (like adiabatic theorems), if possible.
Thanks in advance.