The one-dimensional viscous conservation law is considered on the whole line $$ u_t + f(u)_x=eps u_{xx},quad (x,t)inRRtimesoverline{RP},quad eps>0, $$ subject to positive measure initial data. The flux $fin C^1(RR)$ 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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