Is there a Gronwall-type inequality for bounding $u(t)$ such that $$\vert \partial_t u(t)\vert\leq C \{ u(t)+u(t)^\alpha\}$$ with $\alpha>1$ ?
2026-02-23 06:52:27.1771829547
Gronwall type inequality
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To avoid complications I will assume that $u,\partial_tu\ge0$. Then we have $$ \frac{\partial u_t}{u+u^\alpha}\le C. $$ Let $F(u)=\int_0^uds/(s+s^\alpha)$. Then $$ F(u(t))\le C(t-t_0)+F(u(t_0)),\quad t\ge t_0. $$ and $$ u(t)\le F^{-1}\bigl(C(t-t_0)+F(u(t_0))\bigr),\quad t\ge t_0. $$ Observe that since $\alpha>1$ $F^{-1}$ will be defined on a finite domain.