Consider the heat equation
$$ u_{t} = k(t) \Delta u + f(t,x),~~x \in \mathbb{R}^{n},t>0 $$$$ u(0,x) = u_{0}(x). $$
If $k(t) = k$, then
$$ u(t,x) = \int_{\mathbb{R}^{n}}\phi(x-y,kt)u_{0}(y)dy + \int_{0}^{kt} \int_{\mathbb{R}^{n}}\phi(x-y,kt-\tau)f(y,\tau) d \tau dy $$where$$ \phi(x,t) = \begin{cases} \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{4t}} & t>0 \\ 0 & t<0. \end{cases} $$But if $k$ is time dependent on $t$ (I am assuming $k(t)$ is sufficiently smooth function) then is it possible to find solution?
It might be an easy question, but I am not getting any idea, how to solve it. Any reference would also be welcome.
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