We consider a singular parabolic equation tβut − ∆u = f, for (x,t)∈ Ω × (0,T), arising in symmetric boundary layer flows. Here Ω ⊂ RN is a bounded domain with C2 boundary ∂Ω,β ≤ 1,f = f(t,x) 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 u0 ϵ L2(Ω), 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 tut − ∆u = f. The initial data cannot be arbitrarily chosen. In fact, if f is continuous and f(t) → f0, as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem −∆u = f0, for x ∈ Ω, with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.
The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation
