A BVP nonexistence proof using Green's Theorem

Several recent papers investigate the boundary value problem ϕ^′′(t)+λϕ^′(t)+ϕ(t)²=0,t≥0subject to ϕ(0)=1,ϕ(∞)=0,which arises in certain situations of boundary layer flow. Previous work on the problem established the existence of a λ_min∈[1,2∕3] such that solutions exist for λ≥λ_min. It has been conjectured that for λ<λ_min no solution exists. We improve existing results by proving that for λ<λ₁≈.96105 no solution to the boundary value problem exists. The proof employs a novel application of Green's Theorem and is applicable to other boundary value problems.