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Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations. (English) Zbl 1317.49050
Summary: We show that in one space dimension, Lipschitz solutions of generalized extremal surface equations are equivalent to entropy solutions in \(L^{\infty}(\mathbb{R})\) 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 \(t \to +\infty\) of the entropy solutions to explicit traveling waves in the \(L^1(\mathbb{R})\) 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 \(L^{1}\) stabilities of the entropy solutions.
©2011 American Institute of Physics

MSC:
49Q05 Minimal surfaces and optimization
49K40 Sensitivity, stability, well-posedness
53A05 Surfaces in Euclidean and related spaces
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
35C07 Traveling wave solutions
35L65 Hyperbolic conservation laws
35B35 Stability in context of PDEs
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