On the Cauchy problem for a one-dimensional conservation law with initial conditions coinciding with a power or exponential function at infinity (Q2671938)

The author constructs locally bounded entropy solutions to the Cauchy problem for the scalar conservation law \(u_t+|u|^{\alpha-1}u_x=u_t+(|u|^{\alpha-1}u/\alpha)_x=0\), \(\alpha>1\). The initial function is supposed to be either a power or exponential nonnegative function. The constructed solutions are piecewise smooth and contain a countable family of discontinuity lines going from \(-\infty\). Surprisingly, the constructed solutions change their sign after passing through each discontinuity. Moreover, it is shown that a nonnegative entropy solution does not actually exist.