It is well known that for a harmonic oscillator with linear damping, $\ddot x+c\dot x+x=0$ with positive $c$, the amplitude of the oscillations decays exponentially when $c<2$. If it is higher than $2$, the system fails to oscillate at all and is said to be overdamped.
Suppose the damping is nonlinear instead, following a power law $\ddot x+c\lvert \dot x\rvert^{p-1}\dot x+x=0.$ For example, $p=1$ recovers linear damping, while $p=2$ gives quadratic damping which can model aerodynamic drag. I assume that in general a closed-form solution is not possible due to the presence of the absolute value signs. What can be said about the asymptotic behaviour of the system?
Edit: While @doraemonpaul's comment and @mjqxxx's answer are very nice, I am more interested in stronger results than merely the existence or absence of overdamping. For comparison, consider a first-order nonlinear decay equation, $\dot x+\lvert x\rvert^{p-1}x=0.$ The solution to this has the form $x = \pm(p-1)(t-t_0)^{1/(1-p)}$ with certain conditions on $t_0$. When $p<1$, the solution drops to zero in finite time; when $p>1$, it decays roughly as $t^{-1/(p-1)}$ which is much slower than exponential. What are the corresponding characterizations of how the amplitude of the nonlinearly damped harmonic oscillator behaves? What is the exponent of the decay when $p > 1$? Can the system come to rest in finite time if $p < 1$?