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How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = (q -p) \delta$, where $\delta$ is the Dirac mass in $0$ and $0 <p, q$?

Note: For the case of the Burgers equation $$u_t + (u^2)_x = 0$$ the computation of N-wave solutions is in equation (2.1) of the paper

Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ZBL0545.35057.

But I don't see how to extend this computation to the model above in the question.

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  • $\begingroup$ You need to be careful about your notation. Burger's equation should be written in the form $u_t + (u^2)_x = 0$. What you have written is not this. For the equation that you are asking about, do you mean $u_t - (u^2-u)_x = 0$? $\endgroup$
    – Robert Bryant
    Commented Sep 18, 2022 at 9:59
  • $\begingroup$ @RobertBryant Thanks! Yes, I edited the question to avoid ambiguity in the equation $\endgroup$
    – Riku
    Commented Sep 18, 2022 at 10:59

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