Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$:
$$ \begin{cases} u_t = \begin{cases} \sigma_1^2u_{xx} &\text{if } x>\lambda(t),\\ \sigma_2^2u_{xx} &\text{if } x\leq\lambda(t) \end{cases}\\ u(0,x) = u_0(x). \end{cases}\label{1}\tag{1} $$
The equation basically models a diffusion on $\mathbb{R}$ with 2 media of diffusivity $\sigma_1$ and $\sigma_2$. The border of these 2 media is located at $\lambda(t)$ at time $t\in[0,T]$.
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 ? (Existence, uniqueness, regularity, expression,...)
I would also be very intersted in the case $\lambda$ being constant.
Thank you very much !
Analysis of the equation. Initially I was fooled by the structure of the diffusion coefficient and thought it was similar to a Stefan/moving boundary problem as said by whpowell96 in their comment. However, things are different: indeed the Cauchy problem \eqref{1}, after a careful analysis, appears to be the following one $$ \begin{cases} \dfrac{\partial u (t,x)}{\partial t}= \sigma^2(t,x)\dfrac{\partial^2u(t,x)}{\partial x^2}\\ \\ u(0,x) = u_0(x). \end{cases}\label{1b}\tag{1 bis} $$ where $$ \sigma(t,x) = \begin{cases} \sigma_1 &\text{if } x>\lambda(t),\\ \sigma_2 &\text{if } x\leq\lambda(t) \end{cases} $$ i.e. $$ \sigma(t,x)= H\big(x - \lambda(t)\big)(\sigma_1 -\sigma_2) + \sigma_2, $$ where $H(x)$ is the Heaviside function.
Thus, in the general case, problem \eqref{1} is a "simple" Cauchy problem for a parabolic equation with discontinuous, time-dependent coefficients (when $\lambda(t)\equiv\lambda=\text{const.}$, the case you are very interested to, it is just a parabolic equation with discontinuous time-independent coefficients).
Answer to the reference request question. From now we consider the problem \eqref{1b} as an exact equivalent problem to the original one: then note that
Equations (far) more general than \eqref{1b} satisfying the two conditions above were studied first in [4] (§1, p. 3-18 and §3, p 39-63), under the hypothesis of discontinuity of the coefficients: the Authors allow also for their slow growth as the spatial variable goes to infinity, and this includes the case of bounded discontinuous coefficients, and prove a very general maximum principle under these conditions and an associated uniqueness theorem. Nevertheless the existence theorem (§3, p 39-63) is still proved under the Hölder continuity hypothesis. In [1] (followed by [2] where some properties of the solution under weaker conditions are considered) the same uniqueness result was proved under milder hypotheses on the growth of coefficients but nevertheless allowing discontinuity, along with an existence theorem, this time allowing for discontinuous coefficients.
Notes
References [1], [2] and [4] consider the Cauchy problem general $n$-dimensional parabolic equation: there's no restriction on the dimension of the ambient space.
It is not necessary for $\sigma_1$ and $\sigma_2$ to be constants: the structure of the problem and its solution is the same even if we assume that these are properly defined (bounded) functions of the spatial variable $x\in\Bbb R$.
Despite being perhaps the most comprehensive reference on second order parabolic PDEs, [3] does not deal with parabolic equation of the form under so weak hypotheses on the coefficients: the author require only boundedness and measurability for the coefficients for equations in divergence form, but for the general equation they require smoothness in Hölder sense. Nor it does [5]: this latter reference considers only mixed problems, i.e. problems where the spatial variable is only allowed to vary inside a bounded domain $\Omega\Subset \Bbb R^n$ ($\Subset$ means compactly contained), $n\ge 1$.
Reference
[1] W. Bodanko, "Sur le problème de Cauchy et les problèmes de Fourier pour les équations paraboliques dans un domaine non borne", (French) Annales Polonici Mathematici 18, 79-94 (1966), MR197998, Zbl 0139.05504.
[2] W. Bodanko, "Les propriétés des solutions non négatives de l’équation linéaire normale parabolique", (French), Annales Polonici Mathematici 20, 107-117 (1968), MR227621, Zbl 0156.11204.
[3] Ol'ga Alexandrovna Ladyzhenskaya, Vsevolod Alekseevich Solonnikov, and Nina Nikolaevna Ural’tseva, Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. (English) Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society (AMS), pp. XI+648 (1968), DOI:10.1090/mmono/023, MR241822, Zbl 0174.15403.
[4] Arlen Mikhaĭlovich Il'in, Anatoliĭ Sergeevich Kalashnikov, Olga Arsen'evna Oleĭnik, "Linear equations of the second order of parabolic type", (English. Russian original), Russian Mathematical Surveys 17, No. 3, 1-146 (1962); translation from Uspekhi Matematicheskikh Nauk [N. S.] 17, No. 3(105), 3-146 (1962), MR138888, Zbl 0108.28401.
There's also a new English translation: "Linear second-order partial differential equations of the parabolic type", Journal of Mathematical Sciences, New York 108, No. 4, 435-542 (2002), DOI:10.1023/A:1013156322602, MR1875963, Zbl 0988.35001.
[5] Antonio Maugeri, Dian K. Palagachev, Lubomira G. Softova, Elliptic and parabolic equations with discontinuous coefficients, (English), Mathematical Research 109. Weinheim: Wiley-VCH Verlag, pp. 256 (2000), MR2260015 Zbl 0958.35002.