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Transactions of the American Mathematical Society

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Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation


Author: Atsushi Yoshikawa
Journal: Trans. Amer. Math. Soc. 340 (1993), 103-126
MSC: Primary 35L60; Secondary 35A35, 35B40
DOI: https://doi.org/10.1090/S0002-9947-1993-1208881-X
MathSciNet review: 1208881
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Abstract: It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data $ {\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf{R}}^n}, \eta \ne 0$. Here $ {r_1}(\eta )$ is a characteristic vector, and $ {a_0}(\sigma ,x)$ is a smooth scalar function of compact support. Under the additional requirements that $ n = 2$ or $ 3$ and that $ {a_0}(\sigma ,x)$ have the vanishing mean with respect to $ \sigma $, it is shown that a genuine solution exists in a time interval independent of $ \lambda $, and that the formal solution is asymptotic to the genuine solution as $ \lambda \to \infty $.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1208881-X
Article copyright: © Copyright 1993 American Mathematical Society

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