A priori estimates to $u_t - \Delta u = u^2$

Question

My research is now considering the a priori estimates on the equation $$ \begin{cases}u_t - \Delta u = u \min(u,c) \\ u(0,y) = u(1,y)\\ u_x(0,y) = u_x(1,y)\\ \partial_n u(x,0) = \partial_n u(x,1) = 0 \end{cases}$$ on a 2-dimensional periodic domain $[0, 1]^2$, where $c > 0$ is some fixed given constant. I try to do inner product of the whole equation with $u$: $$ \frac{1}{2} \frac{d}{dt}|u|_{L^2}^2 + \|u\|^2_{H^1} = \langle u\min(u,c) , u \rangle. $$ It seems that I have no way to bound the right hand side in the following form: $$ \bigg|\int u^2\min(u,c)\bigg| \leq \epsilon \|u\|^2_{H^1} + C|u|^2_{L^2} + C,$$ because the integral looks like $\int |u|^3 = |u|_{L^3}^3$ and it has exponent to the 3; while the RHS has only 2. Is there a way to fix it?