The Cauchy Problem for the Biharmonic Heat Equation

The Cauchy problem for a parabolic equation in the cylinder $$C_T=D \times (0,T)$$ over a domain $$D$$ in space $$\mathbb{R}^3$$ with initial data given on a strip on the lateral surface of the cylinder is considered. The strip has the form $$S \times (0,T)$$ , where $$S$$ is an open subset of the boundary of $$D$$ . The problem is ill-posed. Under certain natural constraints on the strip configuration, an explicit formula for the solutions of this problem is obtained.