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.
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