Unicity of solution for a parabolic problem?

Question

How can I show that the parabolic problem

$$ \begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases} $$

has a unique solution? Can I use the maximum principle for the Laplace Equation?

Answer 1

A classical result of uniqueness requires only that $$ \int_{-\infty}^{\infty}|f(x)|dx < \infty \mbox{ (finite heat/energy) } \\ \int_{-\infty}^{\infty}|u(x,t)|dx \le M < \infty,\;\;\; t > 0. $$