A Nonstandard Cauchy Problem for the Heat Equation

We consider the Cauchy problem for the heat equation in a cylinder $\mathcal{C}_T = \mathcal{X} \times (0,T)$ over a domain $\mathcal{X}$ in $\mathbb{R}^n$, with data on a strip lying on the lateral surface. The strip is of the form $S \times (0,T)$, where $S$ is an open subset of the boundary of $\mathcal{X}$. The problem is ill-posed. Under natural restrictions on the configuration of $S$, we derive an explicit formula for solutions of this problem.