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A classification of the behavior of the solutions $f(\cdot,a)$ to theordinary 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 behaviorsof $f(\cdot,a)$: 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 proofof these results relies on a transformation to a first-order ordinarydifferential equation and a monotonicity property with respect to $a>0$. Thisclassification is one step in the description of the dynamics near theextinction time of a diffusive Hamilton-Jacobi equation with critical gradientabsorption and fast diffusion.

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