On the solutions of a boundary value problem arising in free convection with prescribed heat flux

Abstract

For given $a\in \mathbb{R}$, c<0, we are concerned with the solution $f_b^{}$ of the differential equation $f^{\prime\prime\prime}+ff^{\prime\prime}+\g(f^{\prime})=0$, satisfying the initial conditions $f(0)=a$, $f^{\prime }(0)=b$, $f^{\prime \prime }(0)=c<0$, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists $b_{\ast }>0$ such that $f_b^{}$ exists on $[0,+\infty )$ and is such that $f_b^{\prime }(t)\to 0$ as $t\to +\infty$, if and only if $b\ge b_{\ast }$. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.