Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$:
$$ \begin{cases} u_t = \sigma_1^2u_{xx} ~~~~\text{if}~~~~ x>\lambda(t),~\sigma_2^2u_{xx} ~~~~\text{if}~~~~ x\leq\lambda(t)\\ u(0,x) = u_0(x). \end{cases} $$
I tried to find some references for a such heat equation but couldn't find it until now. Does someone have any reference, key words or idea to find more on this topic (general solution) ?
Thank you very much !
