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Source-solutions and asymptotic behavior in conservation laws. (English) Zbl 0545.35057
The authors consider a scalar conservation law $$u_ t+\phi(u)_ x=0$$ in a single space variable. The initial data is taken to be a finite measure $$\mu$$ ; hence the initial condition is stated in the weak form $$\lim_{t\to 0}<\psi(\cdot),u(\cdot,t)>=<\psi(\cdot),d\mu>,$$ for all bounded continuous $$\psi$$. The motivation for this study comes from the fact that the case where $$\mu$$ is the Dirac measure is relevant to the study of the asymptotic behavior of u when $$u(\cdot,0)$$ is taken to be an integrable function. The results are different depending on whether u is non-negative, or not, and whether $$\phi$$ is odd or convex. In fact, if u is negative somewhere, the problem is not well posed in general, for data which are measures. These differences do not occur if the data is a bounded integrable function. This is an interesting paper having lots of new ideas and it opens up a new area for study.
Reviewer: J.Smoller

##### MSC:
 35L65 Hyperbolic conservation laws 35L67 Shocks and singularities for hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
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