So i came across the following inequality:
$$\dfrac{\mathrm{d}}{\mathrm{d} t}\left(\int_{\mathbb{R}^n}u^2\mathrm{d}x\right)\leq K\left(\int_{\mathbb{R}^n}u^2\mathrm{d}x\right)^{\frac{1}{2}}, \,\,\,\,\,\mathrm{in} [0,T).$$
and i'm trying to bound the integral. Is there any variation of Gronwall's inequality that works for that case, i.e., can I somehow conclude that $$ \left(\int_{\mathbb{R}^n}u^2\mathrm{d}x\right) \leq C\exp\left(\int_0^t K\,\mathrm{d}s\right)? $$
Obs: in this case, $u\in H^2(\mathbb{R}^n),$ so there are no problems with the integral itself.