Derivative of an unknown function constrained by a differential relation

Question

Given the differential relation: $$\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial x^2}+u\frac{\partial T}{\partial x}$$ with $$u=K\left ( 1+ \left.\begin{matrix}\frac{\partial T}{\partial x} \end{matrix}\right|_{x=0} \right )$$ where $K$ is a constant larger than $0$, $T(x,t)$ and $u(t)$, and knowing that $T(0,t)=1$, $T(\infty,t)=0$ and $T(x,0)=\exp\left ( -\frac{x^2}{\pi } \right )-x \, \text{erfc}\left ( \frac{x}{\sqrt{\pi }}\right )$ (which implies $u(0)=0$), is it possible to estimate somehow $\left.\begin{matrix} \frac{\partial u}{\partial t}\end{matrix}\right|_{t=0}$, or at least to find some bounds for it? Thanks in advance.