Gronwall's inequality for $\dot{x}(t) \le -c|x(t)|^p$, with $c>0$ and $0 < p \le 1$ fixed constant.

Question

Let $c > 0$ and $0 < p \le 1$ be fixed.

Question. What does Gronwall's inequality say about solutions $t \mapsto x_p(t)$ of the differential inequation $$ \dot{x}(t) \le -c|x(t)|^p,\;\forall t \in \mathbb R. \tag{1} $$

Note (to check!). The the properties (i.e inequalities) for $t \mapsto x_1(t)$ can be obtained from $t \mapsto x_p(t)$ (with $0 < p < 1$), by passing the limit $p \to 1$ and using continuity.