I wonder for a parabolic equation $u_t+(a(t,x)u)_x= u_{xx}$,
if we know that $a(t,x)$ is monotonic decreasing in $x$ with $a(t,-\infty)=C_L, a(t,+\infty)=C_R$, $C_L>C_R\geq 0$, are there results developed to give precise pointwise estimates for $u$? Or long-time behaviors? I bet that $||u||_{L^{\infty}}\leq C/\sqrt{t}$.
