In the following $g$ is a function related with some norms of solution of a certain differential eqution: $g$ be a nonnegative continous (if necessary, it is monotone increasing) function satisfying the inequality $$g(t)\leq C_{1}\varepsilon^2+Cg(t)^{(p+1)/2}$$ for $t\in[t_0, T)$ with $0<t_0<T$, where $p>1$ is a number and and $\varepsilon>0$ is a small number, $C$ and $C_{1}>0$ are constants. Also we know that $g(t_0)\leq C^*\varepsilon^2$ for some $C^*>0$ constant. Then I need to show that $$g(t)\leq 2(C_{1}+C^*)\varepsilon^2$$ for $t\in[t_0, T)$ and this implies $T=+\infty$.
I will be grateful for any help or suggesting any reference.