Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is $$M \equiv\int_{x=-\infty}^\infty u(x) ~ dx$$ where $M>0$.

I want to find the minimum value of $$E[u] = \int_{x=-\infty}^\infty u^3 -u^2 +(u_x)^2 ~ dx$$ and ideally the function $u$ that minimizes $E$.

My conjecture is that it is an elliptic function with compact support, but before I dig deep into the analysis I am wondering if the answer is known.

This question arises in determining the attractors of a degenerate Cahn-Hilliard Equation, $$u_t = - [u(u_{xx} +3u^2 -2u)_x]_x $$ which admits non-negative solutions $u(x,t)$ with compact support. The attractors at large time yields the problem above.