Classification of extinction profiles for a one-dimensional diffusive hamilton-jacobi equation with critical absorption

A classification of the behavior of the solutions $f(\cdot,a)$ to the ordinary differential equation $(|f'|^{p-2} f')' + f - |f'|^{p-1} = 0$ in $(0,\infty)$ with initial condition $f(0,a)=a$ and $f'(0,a)=0$ is provided, according to the value of the parameter $a>0$, the exponent $p$ ranging in $(1,2)$. There is a threshold value $a_*$ which separates different behaviors of $f(\cdot,a)$: if $a>a_*$ then $f(\cdot,a)$ vanishes at least once in $(0,\infty)$ and takes negative values while $f(\cdot,a)$ is positive in $(0,\infty)$ and decays algebraically to zero as $r\to\infty$ if $a\in (0,a_*)$. At the threshold value, $f(\cdot,a_*)$ is also positive in $(0,\infty)$ but decays exponentially fast to zero as $r\to\infty$. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to $a>0$. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton-Jacobi equation with critical gradient absorption and fast diffusion.