 Solution

Let $\overline{T}=T_0+\vartheta (t)$, with $0\le \vartheta \le \Delta T$. Apply the same one-boundary four-thirds convective heat-transfer law coefficient $K$ to both boundary layers. The lower inward heat flux is $K(\Delta T-\vartheta {)}^{4/3}$ and the upper outward heat flux is $K{\vartheta }^{4/3}$. The layer's conservation of energy per unit area givesHere $\rho c_p$ is the volumetric heat capacity. This uses a uniform bulk temperature, negligible boundary layer heat storage, constant depth and material properties, prescribed boundary temperatures, no lateral losses or internal heat generation, and identical upper and lower closure coefficients. It deliberately extrapolates the turbulent closure to $t=0$, as the model requires; an actual impulsive start first has a conductive transient. This is the requested evolution equation, without solving it.