Let $v(t)$ be a differentiable function, and consider a solution to the nonhomogeneous heat conduction problem: A solution $u$ such that \begin{align} &u_t = \frac{1}{2}u_{xx} + v(t)u_x, x > 0\\ &u_x(t, 0) = -2v(t)y(t, 0). \end{align} Say we have an initial condition of $u_0 = f(x) \geq 0$ for $x \geq 0.$
My question is two fold: I would like to use energy estimates but need a priori bounds of the form $$\lim_{M\to\infty}u_x(t, M) \to 0?$$
Can we find some explicit solution which will give regularity of $u,$ and what assumptions would be necessary to place on $f$ and $v?$