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Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations

We show that in one space dimension, Lipschitz solutions of generalized extremal surface equations are equivalent to entropy solutions in L∞(R) of a non-strictly hyperbolic system of conservation laws. We obtain an explicit representation formula and the uniqueness of the entropy solutions to the Cauchy problem of the system. By using this formula, we also obtain the convergence and convergence rates as t → +∞ of the entropy solutions to explicit traveling waves in the L1(R) norm. Moreover, when initial data are constants outside of a finite space interval, the entropy solutions become the explicit traveling waves after a finite time. Finally, we prove L1 stabilities of the entropy solutions.