Gronwall's type inequality

Question

Is there a Gronwall's type inequality or ODE lemma concerning the following inequality $$ \frac{d}{dt}f(t)\leq g(t)f^4(t) + \gamma(t)$$ where $f$, $g$ and $\gamma$ are three continuous non-negative real valued functions whenever $t$ in $(0,T)$. If there exist such lemma or Gronwall like inequality, please which is?

Answer 1

Let $s(t)$ be an exact solution with initial value $s(0)=f(0)+\varepsilon$, $ε>0$ some small constant. Assume that $f(t)< s(t)$ does not hold for all times. Let $\tau$ be the first time so that $f(τ)=s(τ)$. Then $$ 0=s(τ)-f(τ)=ε+\int_0^τg(t)\left(s(t)^4-f(t)^4\right)\,dt\ge ε>0. $$ As this is a contradiction, no such $τ$ exists and thus $f(t)< s(t)$ holds for all $t>0$ where $s(t)$ is finite. By continuity, $f(t)\le s(t)$ also holds for the exact solution $s$ with $ε=0$.

Because of the non-linearity, very likely there will be a pole of $s$ at a finite time, as $$ s'(t)\ge g(t)s(t)^4\implies s(t)^{-3}-s(0)^{-3}\le -3\int_0^tg(s)ds\\ \implies s(t)\ge \frac{s(0)}{\sqrt[3]{1-3s(0)^3\int_0^tg(s)ds}} $$ and if $\int_0^tg(s)ds$ grows over the bound $\frac13s(0)^{-3}$, there is a pole in $s$ before that bound is surpassed. After the pole of $s$, $f$ can still exist but is no longer bounded be the differential inequality.