Uniform Gronwall lemma is very useful when establishing uniform estimate in a priori esimates. **Uniform Gronwall lemma** Let $g,h$ and $y$ be three non-negative locally integrable functions on $[t_0,+\infty)$ such that $$ (*)\quad\quad\f{dy}{dt} \leq gy + h,\quad \forall t\geq t_0,$$ and $$\int_t^{t+r} g(s)ds \leq a_1,\quad\int_t^{t+r} h(s)ds \leq a_2,\quad\int_t^{t+r} y(s)ds \leq a_3,\quad \forall t\geq t_0,$$ where $r,a_1,a_2,a_3$ are positive constants. Then $$y(t+r)\leq (\f{a_3}{r}+a_2)e^{a_1},\quad\forall t\geq t_0.$$ //Proof:// Fix $s_0\geq t_0$, and we classically write Condition (*) as $$\f{d}{dt}(e^{-\int_{s_0}^t g(s) ds } y(t)) \leq e^{-\int_{s_0}^t g(s)ds} h(t),$$ and then integrating from $t$ to $s_0+r$ for $s_0\leq t \leq s_0+r$, we find $$e^{-\int_{s_0}^{s_0+r} g(s) ds } y(s_0+r) - e^{-\int_{s_0}^t g(s) ds } y(t) \leq \int_t^{s_0+r} e^{-\int_{s_0}^{t'} g(s)ds} h(t') dt',$$ Multiplying $e^{\int_{s_0}^t g(s) ds }$, we find $$ e^{\int_{s_0}^t g(s) ds } e^{-a_1} y(s_0+r) - y(t) \leq \int_t^{s_0+r} e^{-\int_{t}^{t'} g(s)ds} h(t') dt'\leq \int_t^{s_0+r} h(t')dt' \leq a_2,$$ We finally integrate from $s_0$ to $s_0+r$ with respect to $t$, and we obtain $$y(s_0+r) r e^{-a_1} - a_3 \leq r a_2,$$ which implies $$y(s_0+r) \leq (\f{a_3}{r}+a_2)e^{a_1}.$$