If you convolve the equation with the heat kernel in both the $t$ and $x$ variables, you get an equation of the form
$$
u = H*f(u).
$$
You can then solve this using a contraction mapping or iteration argument using an appropriate norm on $u$ and for sufficiently small $T$. This will give a solution $u$ that decays at infinity (this will be implied by the norm you use) and is smooth for positive $t$ (assuming that $f$ is a smooth function of $u$). I hope someone can provide a specific reference where this is carried out in detail.