Can Duhamel's Integral used for the following advection-diffusion equation?

Question

I have the following IBVP for advection-diffusion equation.

$$\frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial x^2} + \frac{f(t)}{x} \frac{\partial T}{\partial x}$$

where $$T(x,t=0)=0$$

BCs. $$T(x=0,t)=g(t)$$ $$\lim_{x \to \infty} T(x,t)=0$$

Can Duhamel's integral be used to solve the problem if PDE has time dependent coefficient?

Answer 1

You may have a look at the book of K-J. Engel and R. Nagel on semigroups:

Chapter VI. Semigroups Everywhere, 9. Semigroups for Nonautonomous Cauchy Problems (by Roland Schnaubelt).