On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying $$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$$ and $$\frac{d}{dt}\int v^2 + c_2\int |\nabla v|^2 \leq \frac{c_1}{2}\int |\nabla u|^2$$ where all integrals are over $\Omega$ and $c_1$ and $c_2$ are constants independent of $n$.
Suppose that $u(0)$ and $v(0)$ are in $L^\infty$, and additionally, the mean value of $v(t)$ over $\Omega$ is a constant (independent of time). Note that Poincare's inequality is not available for $u$ (unless you subtract the mean value, of course).
I am wondering if I can find a uniform bound on $u$ and/or $v$ in $L^2(0,T;L^2)$ or a better space independent of $n$ (I want to send $n$ to infinity so am looking for these bounds).
I had hoped that the lower order term in the $u$ inequality would help, since there is no $n$ dependence for the $v$ equation. Of course, one can add the two inequalities to remove the gradient of $u$ on the $v$ inequality, and try to multiply by an integrating factor, however, it does not give me what I want.
