a priori estimate

Question

I'm having trouble deriving a priori estimate for the following PDE.
$ \left\{ \begin{array}{l} \partial_tu-\nu\partial_x^2u+\partial_x(u\int_{t-\tau}^{t}u(s,x)ds)=0 \ \ \ t\geq 0,x \in \mathbb{R}\\ u(\theta,x)=u_0(\theta,x)\ \ \ -\tau\leq\theta\leq 0,x \in \mathbb{R} \end{array} \right.\\ \\ $

I multiplied the equation by $u$ and got following equation.
$ \frac{1}{2}\partial_t||u||_{L^2}^2+\nu||\partial_xu||_{L^2}^2=\int_{\mathbb{R}}u\partial_xu\int_{t-\tau}^{t}u(s,x)dsdx\ -(1) $

Then, the estimate for the right side of (1) is
$ \int_{\mathbb{R}}u\partial_xu\int_{t-\tau}^{t}u(s,x)dsdx\\ \leq \frac{1}{2}||u||_{L^2}^2+\frac{1}{2}||\partial_xu\int_{t-\tau}^{t}u(s,x)ds||_{L^2}^2\\ \leq \frac{1}{2}||u||_{L^2}^2+\frac{1}{2}||\partial_xu||_{L^2}^2||\int_{t-\tau}^{t}u(s,x)ds||_{L^\infty}^2\\ \leq \frac{1}{2}||u||_{L^2}^2+\frac{1}{2}||\partial_xu||_{L^2}^2(\sup_{-\tau\leq s\leq t}||u(s)||_{L^\infty})^2 $
producing $||u||_{L^2}^2$ which is the problem.

Is it possible to estimate (1) with $||u||_{L^\infty},||\partial_xu||_{L^2}^2$ but without $||u||_{L^2}^2$?
Or what kind of equation should I consider to produce $||u||_{L^2}^2$ on the left side?
Thank you in advance.