A nonstandard Cauchy problem for the heat equation

详细

We consider the Cauchy problem for the heat equation in a cylinder C_T = X × (0, T) over a domain X in Rⁿ, with data on a strip lying on the lateral surface. The strip is of the form S × (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem.