Interface dynamics in semilinear wave equations
| Autor: | del Pino, Manuel, Jerrard, Robert, Musso, Monica |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00220-019-03632-z |
| Popis: | We consider the wave equation $\varepsilon^2(-\partial_t^2 + \Delta)u + f(u) = 0$ for $0<\varepsilon\ll 1$, where $f$ is the derivative of a balanced, double-well potential, the model case being $f(u) = u-u^3$. For equations of this form, we construct solutions that exhibit an interface of thickness $O(\varepsilon )$ that separates regions where the solution is $O(\varepsilon^k)$ close to $\pm 1$, and that is close to a timelike hypersurface of vanishing {\em Minkowskian} mean curvature. This provides a Minkowskian analog of the numerous results that connect the Euclidean Allen-Cahn equation and minimal surfaces or the parabolic Allen-Cahn equation and motion by mean curvature. Compared to earlier results of the same character, we develop a new constructive approach that applies to a larger class of nonlinearities and yields much more precise information about the solutions under consideration. Comment: 34 pages |
| Databáze: | arXiv |
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