Given $$u_t=u_{xx} - \frac{x}{2t}u_x - u^3$$ for $x,t>0$, with $u \rightarrow \frac{1}{2\sqrt t}$ as $x \rightarrow \infty$ for fixed any $t>0$, then I want to show that a maximum of $u(x,t)$ must either occur on $\{x=0,t>0\}$ or in the limit for fixed $x \geq 0$ as $t \rightarrow 0$.
I know if a maximum occurs in $x,t>0$ then $u_t=0,u_x=0$, and $u_{xx} \leq 0$ at the maximum, but then I don't know what to do with the $u^3$ term. I also need to show uniqueness, but I think that will be to do with also showing that $u$ has a minimum on $\{x=0,t>0\}$ or in the limit for fixed $x \geq 0$ as $t \rightarrow 0$.
