  The one-dimensional viscous conservation law is considered on the whole line
$$
  u_t + f(u)_x=\eps u_{xx},\quad (x,t)\in\RR\times\overline{\RP},\quad
  \eps>0,
  $$
  subject to positive measure initial data.
  The flux $f\in C^1(\RR)$ is assumed to satisfy a $p-$condition, a weak form
of convexity.
  Existence and uniqueness of solutions is established. The method of proof
relies on sharp decay estimates for viscous Hamilton-Jacobi equations.
