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.
Keyword : singular parabolic equation, degenerate parabolic equations, existence, uniqueness, symmetric boundary layer, regularity
This work is licensed under a Creative Commons Attribution 4.0 International License.
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