Consider the following equation and initial contition $\left\{ \begin{array}{ll} u_t-\frac{1}{2} u_{xx}+2au_x=0, & x\in \mathbb{R},t>0 \\ u(x,0)=u_0(x), & x\in \mathbb{R} \end{array} \right.$ where $u_0(x)$ is odd, monotone increasing and bounded over $\mathbb{R}$.
It is easy to check that u_t-\frac{1}{2}u_{xx}<0. Is it possible to deduce from this the sign of the solution $u(0,t)$ for all time?
Thanks in advance for any insight.