A mathematical model for the use of energy resources: a singular parabolic equation

Abstract

We consider a singular parabolic equation, for , arising in symmetric boundary layer flows. Here is a bounded domain with C2 boundary is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity. There are two different cases depending on β. If β < 1, for any initial data, there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem The initial data cannot be arbitrarily chosen. In fact, if f is continuous and , as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem, for , with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.

Keywords

• singular parabolic equation
• degenerate parabolic equations
• existence
• uniqueness
• symmetric boundary layer
• regularity