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An Initial-Value Problem for a Hyperbolic Equation

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Shock Waves and Reaction—Diffusion Equations

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 258))

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Abstract

We consider the equation for the homogeneous operator P:

$$P(u)\, \equiv \,\sum\limits_{\left| \alpha \right|\, = \,m} {{a_\alpha }{D^\alpha }u\, = \,\phi (x),\,x\, = \,(t,\,\xi )\, \in \,{R_ + }\, \times \,{R^n},} $$
(6.1)

with initial data

$$D_0^j\,u(0,\,\xi )\, = \,{\psi _j}(\xi ),\,0 \leqslant \,j\,{\text{ < }}\,m.$$
(6.2)

We assume that each aα is constant, and that the hyperplane t = 0 is non-characteristic with respect to P. Since the normal vector to the surface t = 0 is (1, 0, …, 0), this means a m …0 ≠ 0; thus we assume a m …0 = 1.

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© 1983 Springer-Verlag New York Inc.

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Smoller, J. (1983). An Initial-Value Problem for a Hyperbolic Equation. In: Shock Waves and Reaction—Diffusion Equations. Grundlehren der mathematischen Wissenschaften, vol 258. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0152-3_6

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  • DOI: https://doi.org/10.1007/978-1-4684-0152-3_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4684-0154-7

  • Online ISBN: 978-1-4684-0152-3

  • eBook Packages: Springer Book Archive

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