| Abstract: |
| We study nonnegative solutions of the Cauchy problem
$$u_t+{\varphi(u)}_x = 0 \mbox{~in~}{\bf R} \times (0,T),~u = u_0 \geq 0 \mbox{~in~}{\bf R} \times {0},$$
where $u_0$ is a Radon measure and $\varphi(u) : [0,\infty) \rightarrow {\bf R}$ is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on $\varphi$, we prove their uniqueness if the singular part of $u_0$ is a finite superposition of Dirac masses. In terms of the behaviour of $\varphi$ at infinity we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive {\em waiting time} (in the linear case $\varphi(u))=u$ this happens for all times). In the latter case we describe the evolution of the singular parts.
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