Laplace transform of Diffusion Equation

$\displaystyle D(t)\frac{\partial^{2}C}{\partial x^{2}}=\frac{\partial C}{\partial t},$

I've been using the Laplace transform to solve the above equation with certain boundary conditions. Ive done this successfully when D is constant. However, I have now figured out that D is infact not constant in the system im looking at, it depends on t. The exact form of D(t) is not known, but I know which value it starts for and that it decreases towards another value. My thinking was to choose a D(t) that made the problem solvable. My initial guess was:

$\displaystyle D(t)=D_{1}+D_{2}(1-e^{\frac{-k}{t}})$

which leads me to figure out

$\displaystyle \mathcal{L}\left\{ D(t)\frac{\partial^{2}C}{\partial x^{2}}\right\}$

or

$\displaystyle

\mathcal{L}\left\{ \frac{1}{D(t)}\frac{\partial C}{\partial t}\right\}$?

Is there any easy way to do this, for example some rules that let me isolate $\displaystyle \mathcal{L}\left\{D(t)\right\} $