Asymptotic behavior of solutions to a BVP from fluid mechanics

In this paper we investigate a boundary value problem (BVP) derived from a model of boundary layer flow past a suddenly heated vertical surface in a saturated porous medium. The surface is heated at a rate proportional to x^κ where x measures distance along the wall and κ > -1 is constant. Previous results have established the existence of a continuum of solutions for -1 < k < -1/2. Here we further analyze this continuum and determine that precisely one solution of this continuum approaches the boundary condition at infinity exponentially while all others approach algebraically. Previous results also showed that the solution to the BVP is unique for -1/2 ≤ κ < 0. Here we extend the range of uniqueness to 0 ≤ κ ≤ 1. Finally, the physical implications of the mathematical results are discussed and a comparison is made to the solutions for the related case of prescribed surface temperature on the surface.